Question

Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.

Answer 1

(a) Let f is continuously differentiable. Then this means Df (derivative of f) exists and is also continuous. So, take any arbitrary a in U. Using the definition of continuity, for any
L
, there exists $\delta >0$ such that for any x in U, whenever

  

a

we have

.

Df(x)-Df(a)|| < e

Since a is arbitrary, so above holds for any a in U. This proves one side.

Now, we move on to prove the other side. For each a in U and for any
L
, there exists $\delta >0$ such that for any x in U, whenever

  

a

we have

Df(x)-Df(a)|| < e

Above means that for any a in U, Df(x) exists in the neighborhood of 'a' and as 'a' is arbitrary, we conclude that f is derivable. Further the given assertion also means that Df is also continuous as whenever x approaches to a, Df(x) approaches to Df(a). So, f is continuously differentiable.