Increasing function on R that is discontinuous on the rationals











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The question (Folland's Real Analysis, 3.5.30) asks to produce an increasing function on $mathbb{R}$ whose set of discontinuities is the rationals. My train of thought is as follows:



Let $f_0$ be the identity. We want to create a jump at every rational number, while keeping the function increasing. Create $f_1$ by cutting $f_0$ at each integer, and halving the slope of each resulting line segment, fixing the left endpoints. This results in a sort of slanted stair that is still increasing and discontinuous exactly at the integers. Given $f_{n-1}$, we create $f_n$ by cutting each segment of $f_{n-1}$ into $n$ parts and halving the slope of each resulting segment, again fixing the left endpoints. Note: this is equivalent to making cuts at all rational points $frac{m}{n!}$ at the $n^text{th}$ step, with $m$ and $n$ not necessarily coprime.



The limit of such sequence of functions exists. It's easy to prove that any rational point is eventually a left endpoint, and is thus kept fixed by all successive functions in the sequence. For an irrational point $x$ we can make two sequences $a_n$ and $b_n$ of rationals converging from the left and right respectively, such that $a_n=frac{m}{n!}<x$ and $b_n=frac{m+1}{n!}>x$. Given $epsilon$, we can then choose $N$ large enough with $a_N$ and $b_N$ on the same segment of $f_{N-1}$ (so the jump from $f_{N-1}(a_N)$ to $f_{N-1}(b_N)$ is small), with this jump being less than $epsilon$. Since we can pinpoint the value of $f_n(x)$ between arbitrarily close values, the limit $f(x)=lim_n f_n(x)$ exists.



It's clear that the function is discontinuous at all rationals since it is constructed to be, and it is increasing since each $f_n$ is increasing.



Can I say that $f$ is continuous at each irrational? My gut feeling is that, since any $delta$-ball around an irrational contains a rational to the left and one to the right, we cannot necessarily say $f$ is continuous there. However we know by theorem 3.23 in the book that the set of discontinuities of an increasing function is countable. Is there a contradiction somewhere? If so, can the construction be tweaked or is a different construction necessary?



Thank you.










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  • 1




    Your construction looks promising. A function can be continuous at $x$ even if every neighbourhood of $x$ contains a point of discontinuity (e.g., let $f(x)$ be $0$ if $x$ is rational and be $x$ if $x$ is irrational, then $f$ is continuous at $0$). Try constructing the $epsilon$-$delta$ proof that your function is continuous at irrational numbers.
    – Rob Arthan
    Nov 21 at 23:55












  • What I'm thinking is this: let $epsilon>0$, assume there exists $delta$ such that in a $delta$-ball around $x$, $|f(x)-f(y)|<epsilon$. Take two such rationals $p$ and $q$; then we have $|f(p)-f(q)|<2epsilon$ by triangle inequality. Is this a problem?
    – Luca S.
    Nov 21 at 23:56












  • I don't think so. Write the $epsilon$-$delta$ argument down and check it for yourself.
    – Rob Arthan
    Nov 22 at 0:01






  • 1




    I upvoted the question since I think your approach will work. Would you be interested in another function having the required properties? [I have one for which the properties are somewhat simple to show.]
    – coffeemath
    Nov 22 at 4:35










  • Sure, go ahead!
    – Luca S.
    Nov 22 at 7:17















up vote
4
down vote

favorite
4












The question (Folland's Real Analysis, 3.5.30) asks to produce an increasing function on $mathbb{R}$ whose set of discontinuities is the rationals. My train of thought is as follows:



Let $f_0$ be the identity. We want to create a jump at every rational number, while keeping the function increasing. Create $f_1$ by cutting $f_0$ at each integer, and halving the slope of each resulting line segment, fixing the left endpoints. This results in a sort of slanted stair that is still increasing and discontinuous exactly at the integers. Given $f_{n-1}$, we create $f_n$ by cutting each segment of $f_{n-1}$ into $n$ parts and halving the slope of each resulting segment, again fixing the left endpoints. Note: this is equivalent to making cuts at all rational points $frac{m}{n!}$ at the $n^text{th}$ step, with $m$ and $n$ not necessarily coprime.



The limit of such sequence of functions exists. It's easy to prove that any rational point is eventually a left endpoint, and is thus kept fixed by all successive functions in the sequence. For an irrational point $x$ we can make two sequences $a_n$ and $b_n$ of rationals converging from the left and right respectively, such that $a_n=frac{m}{n!}<x$ and $b_n=frac{m+1}{n!}>x$. Given $epsilon$, we can then choose $N$ large enough with $a_N$ and $b_N$ on the same segment of $f_{N-1}$ (so the jump from $f_{N-1}(a_N)$ to $f_{N-1}(b_N)$ is small), with this jump being less than $epsilon$. Since we can pinpoint the value of $f_n(x)$ between arbitrarily close values, the limit $f(x)=lim_n f_n(x)$ exists.



It's clear that the function is discontinuous at all rationals since it is constructed to be, and it is increasing since each $f_n$ is increasing.



Can I say that $f$ is continuous at each irrational? My gut feeling is that, since any $delta$-ball around an irrational contains a rational to the left and one to the right, we cannot necessarily say $f$ is continuous there. However we know by theorem 3.23 in the book that the set of discontinuities of an increasing function is countable. Is there a contradiction somewhere? If so, can the construction be tweaked or is a different construction necessary?



Thank you.










share|cite|improve this question


















  • 1




    Your construction looks promising. A function can be continuous at $x$ even if every neighbourhood of $x$ contains a point of discontinuity (e.g., let $f(x)$ be $0$ if $x$ is rational and be $x$ if $x$ is irrational, then $f$ is continuous at $0$). Try constructing the $epsilon$-$delta$ proof that your function is continuous at irrational numbers.
    – Rob Arthan
    Nov 21 at 23:55












  • What I'm thinking is this: let $epsilon>0$, assume there exists $delta$ such that in a $delta$-ball around $x$, $|f(x)-f(y)|<epsilon$. Take two such rationals $p$ and $q$; then we have $|f(p)-f(q)|<2epsilon$ by triangle inequality. Is this a problem?
    – Luca S.
    Nov 21 at 23:56












  • I don't think so. Write the $epsilon$-$delta$ argument down and check it for yourself.
    – Rob Arthan
    Nov 22 at 0:01






  • 1




    I upvoted the question since I think your approach will work. Would you be interested in another function having the required properties? [I have one for which the properties are somewhat simple to show.]
    – coffeemath
    Nov 22 at 4:35










  • Sure, go ahead!
    – Luca S.
    Nov 22 at 7:17













up vote
4
down vote

favorite
4









up vote
4
down vote

favorite
4






4





The question (Folland's Real Analysis, 3.5.30) asks to produce an increasing function on $mathbb{R}$ whose set of discontinuities is the rationals. My train of thought is as follows:



Let $f_0$ be the identity. We want to create a jump at every rational number, while keeping the function increasing. Create $f_1$ by cutting $f_0$ at each integer, and halving the slope of each resulting line segment, fixing the left endpoints. This results in a sort of slanted stair that is still increasing and discontinuous exactly at the integers. Given $f_{n-1}$, we create $f_n$ by cutting each segment of $f_{n-1}$ into $n$ parts and halving the slope of each resulting segment, again fixing the left endpoints. Note: this is equivalent to making cuts at all rational points $frac{m}{n!}$ at the $n^text{th}$ step, with $m$ and $n$ not necessarily coprime.



The limit of such sequence of functions exists. It's easy to prove that any rational point is eventually a left endpoint, and is thus kept fixed by all successive functions in the sequence. For an irrational point $x$ we can make two sequences $a_n$ and $b_n$ of rationals converging from the left and right respectively, such that $a_n=frac{m}{n!}<x$ and $b_n=frac{m+1}{n!}>x$. Given $epsilon$, we can then choose $N$ large enough with $a_N$ and $b_N$ on the same segment of $f_{N-1}$ (so the jump from $f_{N-1}(a_N)$ to $f_{N-1}(b_N)$ is small), with this jump being less than $epsilon$. Since we can pinpoint the value of $f_n(x)$ between arbitrarily close values, the limit $f(x)=lim_n f_n(x)$ exists.



It's clear that the function is discontinuous at all rationals since it is constructed to be, and it is increasing since each $f_n$ is increasing.



Can I say that $f$ is continuous at each irrational? My gut feeling is that, since any $delta$-ball around an irrational contains a rational to the left and one to the right, we cannot necessarily say $f$ is continuous there. However we know by theorem 3.23 in the book that the set of discontinuities of an increasing function is countable. Is there a contradiction somewhere? If so, can the construction be tweaked or is a different construction necessary?



Thank you.










share|cite|improve this question













The question (Folland's Real Analysis, 3.5.30) asks to produce an increasing function on $mathbb{R}$ whose set of discontinuities is the rationals. My train of thought is as follows:



Let $f_0$ be the identity. We want to create a jump at every rational number, while keeping the function increasing. Create $f_1$ by cutting $f_0$ at each integer, and halving the slope of each resulting line segment, fixing the left endpoints. This results in a sort of slanted stair that is still increasing and discontinuous exactly at the integers. Given $f_{n-1}$, we create $f_n$ by cutting each segment of $f_{n-1}$ into $n$ parts and halving the slope of each resulting segment, again fixing the left endpoints. Note: this is equivalent to making cuts at all rational points $frac{m}{n!}$ at the $n^text{th}$ step, with $m$ and $n$ not necessarily coprime.



The limit of such sequence of functions exists. It's easy to prove that any rational point is eventually a left endpoint, and is thus kept fixed by all successive functions in the sequence. For an irrational point $x$ we can make two sequences $a_n$ and $b_n$ of rationals converging from the left and right respectively, such that $a_n=frac{m}{n!}<x$ and $b_n=frac{m+1}{n!}>x$. Given $epsilon$, we can then choose $N$ large enough with $a_N$ and $b_N$ on the same segment of $f_{N-1}$ (so the jump from $f_{N-1}(a_N)$ to $f_{N-1}(b_N)$ is small), with this jump being less than $epsilon$. Since we can pinpoint the value of $f_n(x)$ between arbitrarily close values, the limit $f(x)=lim_n f_n(x)$ exists.



It's clear that the function is discontinuous at all rationals since it is constructed to be, and it is increasing since each $f_n$ is increasing.



Can I say that $f$ is continuous at each irrational? My gut feeling is that, since any $delta$-ball around an irrational contains a rational to the left and one to the right, we cannot necessarily say $f$ is continuous there. However we know by theorem 3.23 in the book that the set of discontinuities of an increasing function is countable. Is there a contradiction somewhere? If so, can the construction be tweaked or is a different construction necessary?



Thank you.







real-analysis






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share|cite|improve this question











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asked Nov 21 at 23:42









Luca S.

404




404








  • 1




    Your construction looks promising. A function can be continuous at $x$ even if every neighbourhood of $x$ contains a point of discontinuity (e.g., let $f(x)$ be $0$ if $x$ is rational and be $x$ if $x$ is irrational, then $f$ is continuous at $0$). Try constructing the $epsilon$-$delta$ proof that your function is continuous at irrational numbers.
    – Rob Arthan
    Nov 21 at 23:55












  • What I'm thinking is this: let $epsilon>0$, assume there exists $delta$ such that in a $delta$-ball around $x$, $|f(x)-f(y)|<epsilon$. Take two such rationals $p$ and $q$; then we have $|f(p)-f(q)|<2epsilon$ by triangle inequality. Is this a problem?
    – Luca S.
    Nov 21 at 23:56












  • I don't think so. Write the $epsilon$-$delta$ argument down and check it for yourself.
    – Rob Arthan
    Nov 22 at 0:01






  • 1




    I upvoted the question since I think your approach will work. Would you be interested in another function having the required properties? [I have one for which the properties are somewhat simple to show.]
    – coffeemath
    Nov 22 at 4:35










  • Sure, go ahead!
    – Luca S.
    Nov 22 at 7:17














  • 1




    Your construction looks promising. A function can be continuous at $x$ even if every neighbourhood of $x$ contains a point of discontinuity (e.g., let $f(x)$ be $0$ if $x$ is rational and be $x$ if $x$ is irrational, then $f$ is continuous at $0$). Try constructing the $epsilon$-$delta$ proof that your function is continuous at irrational numbers.
    – Rob Arthan
    Nov 21 at 23:55












  • What I'm thinking is this: let $epsilon>0$, assume there exists $delta$ such that in a $delta$-ball around $x$, $|f(x)-f(y)|<epsilon$. Take two such rationals $p$ and $q$; then we have $|f(p)-f(q)|<2epsilon$ by triangle inequality. Is this a problem?
    – Luca S.
    Nov 21 at 23:56












  • I don't think so. Write the $epsilon$-$delta$ argument down and check it for yourself.
    – Rob Arthan
    Nov 22 at 0:01






  • 1




    I upvoted the question since I think your approach will work. Would you be interested in another function having the required properties? [I have one for which the properties are somewhat simple to show.]
    – coffeemath
    Nov 22 at 4:35










  • Sure, go ahead!
    – Luca S.
    Nov 22 at 7:17








1




1




Your construction looks promising. A function can be continuous at $x$ even if every neighbourhood of $x$ contains a point of discontinuity (e.g., let $f(x)$ be $0$ if $x$ is rational and be $x$ if $x$ is irrational, then $f$ is continuous at $0$). Try constructing the $epsilon$-$delta$ proof that your function is continuous at irrational numbers.
– Rob Arthan
Nov 21 at 23:55






Your construction looks promising. A function can be continuous at $x$ even if every neighbourhood of $x$ contains a point of discontinuity (e.g., let $f(x)$ be $0$ if $x$ is rational and be $x$ if $x$ is irrational, then $f$ is continuous at $0$). Try constructing the $epsilon$-$delta$ proof that your function is continuous at irrational numbers.
– Rob Arthan
Nov 21 at 23:55














What I'm thinking is this: let $epsilon>0$, assume there exists $delta$ such that in a $delta$-ball around $x$, $|f(x)-f(y)|<epsilon$. Take two such rationals $p$ and $q$; then we have $|f(p)-f(q)|<2epsilon$ by triangle inequality. Is this a problem?
– Luca S.
Nov 21 at 23:56






What I'm thinking is this: let $epsilon>0$, assume there exists $delta$ such that in a $delta$-ball around $x$, $|f(x)-f(y)|<epsilon$. Take two such rationals $p$ and $q$; then we have $|f(p)-f(q)|<2epsilon$ by triangle inequality. Is this a problem?
– Luca S.
Nov 21 at 23:56














I don't think so. Write the $epsilon$-$delta$ argument down and check it for yourself.
– Rob Arthan
Nov 22 at 0:01




I don't think so. Write the $epsilon$-$delta$ argument down and check it for yourself.
– Rob Arthan
Nov 22 at 0:01




1




1




I upvoted the question since I think your approach will work. Would you be interested in another function having the required properties? [I have one for which the properties are somewhat simple to show.]
– coffeemath
Nov 22 at 4:35




I upvoted the question since I think your approach will work. Would you be interested in another function having the required properties? [I have one for which the properties are somewhat simple to show.]
– coffeemath
Nov 22 at 4:35












Sure, go ahead!
– Luca S.
Nov 22 at 7:17




Sure, go ahead!
– Luca S.
Nov 22 at 7:17










1 Answer
1






active

oldest

votes

















up vote
4
down vote



accepted










There is a simpler construction:



If we take a numeration $(q_n)_{n in mathbb{N}}$ of $mathbb{Q}$, we can define
$$f(x) := sum_{k=1}^infty frac{1}{2^k} 1_{[q_k, infty)}(x).$$
The convergence is uniform and this function is monotone increasing by construction. Since all $1_{[q_n,infty)}$ are continuous in all points of $mathbb{R} setminus mathbb{Q}$, the limes is this too. (If $x$ is irrational, then we can take $varepsilon >0$ so small that $(x-varepsilon,x+varepsilon)$ doesn't contain the points $q_1,ldots,q_n$. So $|f(x)-f(y)| le sum_{k=n}^infty 2^{-k} le 2^{1-n}$.)



Fix $n$. For any $max_{q_i < q_n,i=1,ldots n} q_i < x < q_n < y < max_{q_i > q_n,i =1 ldots, n}$ q_i we have
$f(x) + 2^{-n} le f(y)$. Thus $f$ is discontinuous in any point of $mathbb{Q}$.






share|cite|improve this answer





















  • I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
    – Rchn
    Nov 22 at 10:38










  • In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
    – p4sch
    Nov 22 at 20:49













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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
4
down vote



accepted










There is a simpler construction:



If we take a numeration $(q_n)_{n in mathbb{N}}$ of $mathbb{Q}$, we can define
$$f(x) := sum_{k=1}^infty frac{1}{2^k} 1_{[q_k, infty)}(x).$$
The convergence is uniform and this function is monotone increasing by construction. Since all $1_{[q_n,infty)}$ are continuous in all points of $mathbb{R} setminus mathbb{Q}$, the limes is this too. (If $x$ is irrational, then we can take $varepsilon >0$ so small that $(x-varepsilon,x+varepsilon)$ doesn't contain the points $q_1,ldots,q_n$. So $|f(x)-f(y)| le sum_{k=n}^infty 2^{-k} le 2^{1-n}$.)



Fix $n$. For any $max_{q_i < q_n,i=1,ldots n} q_i < x < q_n < y < max_{q_i > q_n,i =1 ldots, n}$ q_i we have
$f(x) + 2^{-n} le f(y)$. Thus $f$ is discontinuous in any point of $mathbb{Q}$.






share|cite|improve this answer





















  • I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
    – Rchn
    Nov 22 at 10:38










  • In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
    – p4sch
    Nov 22 at 20:49

















up vote
4
down vote



accepted










There is a simpler construction:



If we take a numeration $(q_n)_{n in mathbb{N}}$ of $mathbb{Q}$, we can define
$$f(x) := sum_{k=1}^infty frac{1}{2^k} 1_{[q_k, infty)}(x).$$
The convergence is uniform and this function is monotone increasing by construction. Since all $1_{[q_n,infty)}$ are continuous in all points of $mathbb{R} setminus mathbb{Q}$, the limes is this too. (If $x$ is irrational, then we can take $varepsilon >0$ so small that $(x-varepsilon,x+varepsilon)$ doesn't contain the points $q_1,ldots,q_n$. So $|f(x)-f(y)| le sum_{k=n}^infty 2^{-k} le 2^{1-n}$.)



Fix $n$. For any $max_{q_i < q_n,i=1,ldots n} q_i < x < q_n < y < max_{q_i > q_n,i =1 ldots, n}$ q_i we have
$f(x) + 2^{-n} le f(y)$. Thus $f$ is discontinuous in any point of $mathbb{Q}$.






share|cite|improve this answer





















  • I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
    – Rchn
    Nov 22 at 10:38










  • In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
    – p4sch
    Nov 22 at 20:49















up vote
4
down vote



accepted







up vote
4
down vote



accepted






There is a simpler construction:



If we take a numeration $(q_n)_{n in mathbb{N}}$ of $mathbb{Q}$, we can define
$$f(x) := sum_{k=1}^infty frac{1}{2^k} 1_{[q_k, infty)}(x).$$
The convergence is uniform and this function is monotone increasing by construction. Since all $1_{[q_n,infty)}$ are continuous in all points of $mathbb{R} setminus mathbb{Q}$, the limes is this too. (If $x$ is irrational, then we can take $varepsilon >0$ so small that $(x-varepsilon,x+varepsilon)$ doesn't contain the points $q_1,ldots,q_n$. So $|f(x)-f(y)| le sum_{k=n}^infty 2^{-k} le 2^{1-n}$.)



Fix $n$. For any $max_{q_i < q_n,i=1,ldots n} q_i < x < q_n < y < max_{q_i > q_n,i =1 ldots, n}$ q_i we have
$f(x) + 2^{-n} le f(y)$. Thus $f$ is discontinuous in any point of $mathbb{Q}$.






share|cite|improve this answer












There is a simpler construction:



If we take a numeration $(q_n)_{n in mathbb{N}}$ of $mathbb{Q}$, we can define
$$f(x) := sum_{k=1}^infty frac{1}{2^k} 1_{[q_k, infty)}(x).$$
The convergence is uniform and this function is monotone increasing by construction. Since all $1_{[q_n,infty)}$ are continuous in all points of $mathbb{R} setminus mathbb{Q}$, the limes is this too. (If $x$ is irrational, then we can take $varepsilon >0$ so small that $(x-varepsilon,x+varepsilon)$ doesn't contain the points $q_1,ldots,q_n$. So $|f(x)-f(y)| le sum_{k=n}^infty 2^{-k} le 2^{1-n}$.)



Fix $n$. For any $max_{q_i < q_n,i=1,ldots n} q_i < x < q_n < y < max_{q_i > q_n,i =1 ldots, n}$ q_i we have
$f(x) + 2^{-n} le f(y)$. Thus $f$ is discontinuous in any point of $mathbb{Q}$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 22 at 10:09









p4sch

4,800217




4,800217












  • I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
    – Rchn
    Nov 22 at 10:38










  • In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
    – p4sch
    Nov 22 at 20:49




















  • I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
    – Rchn
    Nov 22 at 10:38










  • In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
    – p4sch
    Nov 22 at 20:49


















I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
– Rchn
Nov 22 at 10:38




I'm having such a hard time visualizing a function constructed like this. The proof is sound though.
– Rchn
Nov 22 at 10:38












In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
– p4sch
Nov 22 at 20:49






In fact, you cannot really visualize this function. The key point is that we only add a jump of height $2^{-n}$ in every step. Define $g(x) = sum_{k=1}^infty frac{1}{2^k} 1_{[1/k,infty)}$. This function has in $x=0$ the same behaviour (and can be visualized): In every neighboorhoud $(-varepsilon,varepsilon)$ we have infinite many jump-discontinuities. However, $g$ is continuous in $x=0$!
– p4sch
Nov 22 at 20:49




















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