# Thread: On the continuity of vectorial function

1. ## 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

2. 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.