# continuity single and multivariable function simple question

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

$$\left|\frac{xy^2}{x^2+y^2}\right|\le\frac{|x|(x^2+y^2)}{x^2+y^2}=|x|\xrightarrow{(x,y)\to(0,0)}0=f(0,0)$$ and for the second function we have
$$\lim_{x\to10^-}f(x)=2\ne f(10)=5$$