Proving differentiability for a function from the definition

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

Please see below

For this problem,

The solution is,

However, does someone please know why we allowed to assume that the derivative exists for f i.e $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$?

Thanks!

 

[fresh_42]

Homework Statement: Please see below
Relevant Equations: Please see below

For this problem,
View attachment 345049
The solution is,
View attachment 345050
However, does someone please know why we allowed to assume that the derivative exists for f i.e $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$?

Thanks!

Because of the first sentence: Let $f$ be a differentiable function.