If the first derivative of a piece wisely defined function is continuos at breaking point, can I write it as a single function? > Let $f: \mathbb{R}^p \to \mathbb{R}^q$ be a piece wisely defined function, defined as > > $$f(x,y)= \begin{cases} g(x,y) & ;(x,y) \not = (0,0) \\\ 0 & ;(x,y)=(0,0) \\\ \end{cases}.$$ Normally, I would have to write $f_x$ again piece wisely, but if I know that $f_x$ is continuous at $(0,0)$, can I write as a single(not-piecewisely-defined) function ?

If a single expression is able to represent a function over a whole domain, there is no need to decompose it piecewise, whatever it is.