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May 22nd, 2017, 10:17 AM  #1 
Newbie Joined: May 2017 From: usa Posts: 1 Thanks: 0  Continuity of vectorial function
Hi every one. Let $\displaystyle f:\mathbb{R}^{*}_{+}\to\mathbb{R}$ a given function and $\displaystyle g:\mathbb{R}^2\to\mathbb{R}^2$ the function defined by: $\displaystyle 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 $\displaystyle f$ for: 1) The continuity of $\displaystyle g$ in $\displaystyle \mathbb{R}^2$? 2) The differentiability of $\displaystyle g$ in $\displaystyle \mathbb{R}^2$? 3) $\displaystyle g\in C^1$ Thank you. 

Tags 
class c1, continuity, differentiability, function, vectorial 
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