Is there a function (non piece-wise unlike below) which is discontinuous but has directional derivative at particular point? I have a manual that says the function has directional derivative at $(0,0)$ but is not continuous at $(0,0)$. $$f(x,y) = \begin{cases} \frac{xy^2}{x^2+y^4} & \text{ if } x \neq 0\\ 0 & \text{ if } x= 0 \end{cases}$$
Can anyone give me few examples which is not defined piece wise as above?
Answer
$$f(x,y)=\lim_{u\to0}\frac{xy^2+u^2}{x^2+y^4+u^2}$$
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