Why $$f(x,y) =\begin{cases} \frac{xy^2}{x^2 +y^2} \mbox{ for } (x,y)\neq (0,0) \\ 0 \mbox{ for } (x,y)= (0,0)\end{cases}$$ is continuous and

$$f(x) =\begin{cases} 2 \mbox{ for } 0>=x>10 \\ 5 \mbox{ for } x>=10\end{cases}$$ seem to be discontinuity?

Or am I wrong?

This is related to my previous question continuity single variable function and multivariable funtion and its parcial derivatives



We have

$$\left|\frac{xy^2}{x^2+y^2}\right|\le\frac{|x|(x^2+y^2)}{x^2+y^2}=|x|\xrightarrow{(x,y)\to(0,0)}0=f(0,0)$$ and for the second function we have

$$\lim_{x\to10^-}f(x)=2\ne f(10)=5$$

Can you take it from this?

  • $\begingroup$ Yes thank you I understood $\endgroup$ – studentNk Aug 26 '14 at 17:13
  • $\begingroup$ You're welcome. $\endgroup$ – user63181 Aug 26 '14 at 17:13

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