
New Member
On the continuity of vectorial function
Hi every one. Let $f:\mathbb{R}^{*}_{+}\to\mathbb{R}$
a given function and $g:\mathbb{R}^2\to\mathbb{R}^2$ the function defined by:
$$g(x,y)=\begin{cases}
(x,y) & \text{if $x\leq 0$}\\
(x,y+f(x)) & \text{if $x>0$}
\end{cases}$$
Questions: What is the condition on $f$ for:
1) The continuity of $g$ in $\mathbb{R}^2$?
2) The differentiability of $g$ in $\mathbb{R}^2$? Thank you

Elite Member
1) g is continuous at a point, (x, y), if and only if f is continuous at that x.
2) g is differentiable at a point, (x, y), if and only if f is differentiable at that x.
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