Lower semicontinuous function is pointwise limit of continuous functions
$begingroup$
I have a function $f:mathbb{R} to mathbb{R}$ that is lower semicontinuous, i.e.,
$$liminf_{y to x} f(y) geq f(x)$$
How do you prove there is an increasing ($f_{n+1} geq f_n$) sequence ${f_n}_{n=1}^infty$ of continuous functions such that $lim_{n to infty} f_n(x) = f(x)$ for all $x$?
This was used in a proof I was reading without any explanation.
real-analysis
asked Dec 3 '18 at 18:20
SASSAS
397210
$endgroup$
add a comment |
$begingroup$
I have a function $f:mathbb{R} to mathbb{R}$ that is lower semicontinuous, i.e.,
$$liminf_{y to x} f(y) geq f(x)$$
How do you prove there is an increasing ($f_{n+1} geq f_n$) sequence ${f_n}_{n=1}^infty$ of continuous functions such that $lim_{n to infty} f_n(x) = f(x)$ for all $x$?
This was used in a proof I was reading without any explanation.
real-analysis
asked Dec 3 '18 at 18:20
SASSAS
397210
$endgroup$
add a comment |
$begingroup$
I have a function $f:mathbb{R} to mathbb{R}$ that is lower semicontinuous, i.e.,
$$liminf_{y to x} f(y) geq f(x)$$
How do you prove there is an increasing ($f_{n+1} geq f_n$) sequence ${f_n}_{n=1}^infty$ of continuous functions such that $lim_{n to infty} f_n(x) = f(x)$ for all $x$?
This was used in a proof I was reading without any explanation.
real-analysis
asked Dec 3 '18 at 18:20
SASSAS
397210
$endgroup$
I have a function $f:mathbb{R} to mathbb{R}$ that is lower semicontinuous, i.e.,
$$liminf_{y to x} f(y) geq f(x)$$
How do you prove there is an increasing ($f_{n+1} geq f_n$) sequence ${f_n}_{n=1}^infty$ of continuous functions such that $lim_{n to infty} f_n(x) = f(x)$ for all $x$?
This was used in a proof I was reading without any explanation.
real-analysis
real-analysis
asked Dec 3 '18 at 18:20
SASSAS
397210
asked Dec 3 '18 at 18:20
SASSAS
397210
asked Dec 3 '18 at 18:20
SASSAS
397210
asked Dec 3 '18 at 18:20
SASSAS
397210
asked Dec 3 '18 at 18:20
SASSAS
397210
397210
add a comment |
add a comment |
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