About function of several variables












1












$begingroup$


Proof f(x,y)=$frac{x^2y}{x^4+y^2},$when $x^2+y^2ne 0$



f(x,y)=0,when $x^2+y^2=0$
is Continuity on (0,0) on half-line x=$t costheta,y=tsintheta ,0le tle+propto$



My attempt ：



f(0,y)=0,f(x,0)=0, so it’s continuity on x-axis,y-axis,



let k=$tan theta$ ,then f=$frac{kx^3}{x^4+k^2x^2}$
= $frac{kx}{x^2+k^2}$
$lim_{k






real-analysis calculus sequences-and-series limits analysis







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    Proof f(x,y)=$frac{x^2y}{x^4+y^2},$when $x^2+y^2ne 0$



    f(x,y)=0,when $x^2+y^2=0$
    is Continuity on (0,0) on half-line x=$t costheta,y=tsintheta ,0le tle+propto$



    My attempt ：



    f(0,y)=0,f(x,0)=0, so it’s continuity on x-axis,y-axis,



    let k=$tan theta$ ,then f=$frac{kx^3}{x^4+k^2x^2}$
    = $frac{kx}{x^2+k^2}$
    $lim_{k






    real-analysis calculus sequences-and-series limits analysis







    share|cite|improve this question











    $endgroup$















      1












      1








      1


      1



      $begingroup$


      Proof f(x,y)=$frac{x^2y}{x^4+y^2},$when $x^2+y^2ne 0$



      f(x,y)=0,when $x^2+y^2=0$
      is Continuity on (0,0) on half-line x=$t costheta,y=tsintheta ,0le tle+propto$



      My attempt ：



      f(0,y)=0,f(x,0)=0, so it’s continuity on x-axis,y-axis,



      let k=$tan theta$ ,then f=$frac{kx^3}{x^4+k^2x^2}$
      = $frac{kx}{x^2+k^2}$
      $lim_{k






      real-analysis calculus sequences-and-series limits analysis







      share|cite|improve this question











      $endgroup$




      Proof f(x,y)=$frac{x^2y}{x^4+y^2},$when $x^2+y^2ne 0$



      f(x,y)=0,when $x^2+y^2=0$
      is Continuity on (0,0) on half-line x=$t costheta,y=tsintheta ,0le tle+propto$



      My attempt ：



      f(0,y)=0,f(x,0)=0, so it’s continuity on x-axis,y-axis,



      let k=$tan theta$ ,then f=$frac{kx^3}{x^4+k^2x^2}$
      = $frac{kx}{x^2+k^2}$
      $lim_{k






      real-analysis calculus sequences-and-series limits analysis




      real-analysis calculus sequences-and-series limits analysis






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 8 at 11:10







      jackson

















      asked Jan 1 at 15:30









      jacksonjackson

      1209




      1209






















          1 Answer
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          $begingroup$

          If $k ne 0$, then $lim_{x to 0^+} frac{kx}{x^2+k^2}=0$ and $lim_{x to 0^-} frac{kx}{x^2+k^2}=0$



          Now, let's consider the $x$-axis, that is when $y=0$ and along that line, $f(x,y)=0$ .



          Hence, it is continuouson those half-lines.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            So I almost right?
            $endgroup$
            – jackson
            Jan 1 at 15:39










          • $begingroup$
            is there any reason that stop you from taking the limit?
            $endgroup$
            – Siong Thye Goh
            Jan 1 at 15:40










          • $begingroup$
            Seems like no reason ...thanks
            $endgroup$
            – jackson
            Jan 1 at 15:42










          • $begingroup$
            But can is proper I use k to replace $theta$
            $endgroup$
            – jackson
            Jan 1 at 15:46










          • $begingroup$
            What about y-axis
            $endgroup$
            – jackson
            Jan 1 at 15:47











          Your Answer





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          1 Answer
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          active

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          3












          $begingroup$

          If $k ne 0$, then $lim_{x to 0^+} frac{kx}{x^2+k^2}=0$ and $lim_{x to 0^-} frac{kx}{x^2+k^2}=0$



          Now, let's consider the $x$-axis, that is when $y=0$ and along that line, $f(x,y)=0$ .



          Hence, it is continuouson those half-lines.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            So I almost right?
            $endgroup$
            – jackson
            Jan 1 at 15:39










          • $begingroup$
            is there any reason that stop you from taking the limit?
            $endgroup$
            – Siong Thye Goh
            Jan 1 at 15:40










          • $begingroup$
            Seems like no reason ...thanks
            $endgroup$
            – jackson
            Jan 1 at 15:42










          • $begingroup$
            But can is proper I use k to replace $theta$
            $endgroup$
            – jackson
            Jan 1 at 15:46










          • $begingroup$
            What about y-axis
            $endgroup$
            – jackson
            Jan 1 at 15:47
















          3












          $begingroup$

          If $k ne 0$, then $lim_{x to 0^+} frac{kx}{x^2+k^2}=0$ and $lim_{x to 0^-} frac{kx}{x^2+k^2}=0$



          Now, let's consider the $x$-axis, that is when $y=0$ and along that line, $f(x,y)=0$ .



          Hence, it is continuouson those half-lines.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            So I almost right?
            $endgroup$
            – jackson
            Jan 1 at 15:39










          • $begingroup$
            is there any reason that stop you from taking the limit?
            $endgroup$
            – Siong Thye Goh
            Jan 1 at 15:40










          • $begingroup$
            Seems like no reason ...thanks
            $endgroup$
            – jackson
            Jan 1 at 15:42










          • $begingroup$
            But can is proper I use k to replace $theta$
            $endgroup$
            – jackson
            Jan 1 at 15:46










          • $begingroup$
            What about y-axis
            $endgroup$
            – jackson
            Jan 1 at 15:47














          3












          3








          3





          $begingroup$

          If $k ne 0$, then $lim_{x to 0^+} frac{kx}{x^2+k^2}=0$ and $lim_{x to 0^-} frac{kx}{x^2+k^2}=0$



          Now, let's consider the $x$-axis, that is when $y=0$ and along that line, $f(x,y)=0$ .



          Hence, it is continuouson those half-lines.






          share|cite|improve this answer











          $endgroup$



          If $k ne 0$, then $lim_{x to 0^+} frac{kx}{x^2+k^2}=0$ and $lim_{x to 0^-} frac{kx}{x^2+k^2}=0$



          Now, let's consider the $x$-axis, that is when $y=0$ and along that line, $f(x,y)=0$ .



          Hence, it is continuouson those half-lines.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 1 at 16:01

























          answered Jan 1 at 15:37









          Siong Thye GohSiong Thye Goh

          102k1466118




          102k1466118












          • $begingroup$
            So I almost right?
            $endgroup$
            – jackson
            Jan 1 at 15:39










          • $begingroup$
            is there any reason that stop you from taking the limit?
            $endgroup$
            – Siong Thye Goh
            Jan 1 at 15:40










          • $begingroup$
            Seems like no reason ...thanks
            $endgroup$
            – jackson
            Jan 1 at 15:42










          • $begingroup$
            But can is proper I use k to replace $theta$
            $endgroup$
            – jackson
            Jan 1 at 15:46










          • $begingroup$
            What about y-axis
            $endgroup$
            – jackson
            Jan 1 at 15:47


















          • $begingroup$
            So I almost right?
            $endgroup$
            – jackson
            Jan 1 at 15:39










          • $begingroup$
            is there any reason that stop you from taking the limit?
            $endgroup$
            – Siong Thye Goh
            Jan 1 at 15:40










          • $begingroup$
            Seems like no reason ...thanks
            $endgroup$
            – jackson
            Jan 1 at 15:42










          • $begingroup$
            But can is proper I use k to replace $theta$
            $endgroup$
            – jackson
            Jan 1 at 15:46










          • $begingroup$
            What about y-axis
            $endgroup$
            – jackson
            Jan 1 at 15:47
















          $begingroup$
          So I almost right?
          $endgroup$
          – jackson
          Jan 1 at 15:39




          $begingroup$
          So I almost right?
          $endgroup$
          – jackson
          Jan 1 at 15:39












          $begingroup$
          is there any reason that stop you from taking the limit?
          $endgroup$
          – Siong Thye Goh
          Jan 1 at 15:40




          $begingroup$
          is there any reason that stop you from taking the limit?
          $endgroup$
          – Siong Thye Goh
          Jan 1 at 15:40












          $begingroup$
          Seems like no reason ...thanks
          $endgroup$
          – jackson
          Jan 1 at 15:42




          $begingroup$
          Seems like no reason ...thanks
          $endgroup$
          – jackson
          Jan 1 at 15:42












          $begingroup$
          But can is proper I use k to replace $theta$
          $endgroup$
          – jackson
          Jan 1 at 15:46




          $begingroup$
          But can is proper I use k to replace $theta$
          $endgroup$
          – jackson
          Jan 1 at 15:46












          $begingroup$
          What about y-axis
          $endgroup$
          – jackson
          Jan 1 at 15:47




          $begingroup$
          What about y-axis
          $endgroup$
          – jackson
          Jan 1 at 15:47


















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