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Thread: On the continuity of vectorial function

  1. #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. #2
    Elite Member
    Join Date
    Jan 2012
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    4,445
    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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