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May 22nd, 2017, 09:17 AM   #1
Joined: May 2017
From: usa

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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 $x>0$}

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.
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class c1, continuity, differentiability, function, vectorial

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