If $f : \mathbb R^{n} \longrightarrow \mathbb R^{m}$ is differentiable at a point $c \in \mathbb R^{n}$ then it is continuous there.

Asked Dec 28 '16 at 06:35

Active Dec 28 '16 at 06:56

Viewed 59 times

My attempt :

Let $f : \mathbb R^{n} \longrightarrow \mathbb R^{m}$ be differentiable at $x = c \in \mathbb R^{n}$.Then there exists a linear operator $T_{c} : \mathbb R^{n} \longrightarrow \mathbb R^{m}$ such that

$f(c + h) - f(c) = T_{c} (h) + ||h|| E_{c} (h)$ , where $E_{c} (h) \rightarrow 0$ as $h \rightarrow 0$.So taking limit as $h \rightarrow 0$ on both sides of the above equation we obtain

$$\lim _{h \rightarrow 0} [f(c + h) - f(c)] = \lim_{h \rightarrow 0} T_{c} (h)$$.

If we can show that $T_{c}$ is continuous at $0$ then our purpose will be served.

But I find difficulty to show this.Please help me.

Thank you in advance.

edited Dec 28 '16 at 06:56

asked Dec 28 '16 at 06:35

Arnab Chattopadhyay.

2,570

I am sure that this question has been asked several times here on MSE,please do a google search before posting questions! – Arpit Kansal Dec 28 '16 at 06:56
@ArpitKansal I don't find any such which met my purpose. – Arnab Chattopadhyay. Dec 28 '16 at 06:58
Since $|T_c h | \le |T_c| |h |$, it follows that $\lim_{h \to 0} T_c h = 0$. – copper.hat Dec 28 '16 at 06:59
See also this. – Dec 28 '16 at 06:59
1

All linear operators on finite dimensional normed spaces are continuous. – copper.hat Dec 28 '16 at 07:06
Yeh I got it.Thanks to copper.hat and G.Sassatelli. – Arnab Chattopadhyay. Dec 28 '16 at 07:49

If $f : \mathbb R^{n} \longrightarrow \mathbb R^{m}$ is differentiable at a point $c \in \mathbb R^{n}$ then it is continuous there.

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