Proving Derivatives

Question

Let $X$ be a function defined on $\mathbb{R}$ and say that $X$ is a linear map if there exists $A\in\mathbb{R}$ s.t. $\forall v\in\mathbb{R}$, $X(v)=Av$.

Let $a\in\mathbb{R}$ and $f$ a function on an interval centered at $a$, except maybe at $a$. We can say $f$ is a friend at $a$ if there exists $X$ s.t.:

$\lim_{h\to0}=\frac{f(a+h)-f(a)-X(h)}{\mid h\mid}=0$

How do we prove the following claims?

a) If f is differentiable at a then f is a friend at a

b) If f is a friend at a then f is differentiable at a

Answer 1

The following limit

$$\lim_{h\to0}=\frac{f(a+h)-f(a)-X(h)}{h}=0$$

is just the definition of differentiable function and the absolute value for $h$ at the denominator is not essential, indeed for $h>0$

$$\frac{f(a+h)-f(a)-X(h)}{|h|}=\frac{f(a+h)-f(a)-X(h)}{h}\to 0$$

and for $h<0$

$$\frac{f(a+h)-f(a)-X(h)}{|h|}=-\frac{f(a+h)-f(a)-X(h)}{h}\to 0$$

and vice versa.

Answer 2

This is an equivalent definition of derivatives. Take $X(h) = f'(a)h$ and you'll have your result directly. Edit: And also it will be useful to remember that, if $X$ is a linear map in $\mathbb{R}$ then $X(h) = X(1)h$ for all $h \in \mathbb{R}$.