Carathéodory's Theorem (Analysis)

Carathéodory's Theorem: Let $$f : I \subset \mathbb{R} \rightarrow \mathbb{R}$$ is differentiable at $$c \in I$$ if and only if $$\exists \varphi : I \rightarrow \mathbb{R}$$ that is continuous at $$ c $$ and satisfies

$$(1) \quad f(x) - f(c) = \varphi (x)(x-c) \quad \forall x \in I$$

and

$$(2) \quad \varphi (c) = f'(c)$$

Proof
$$( \Rightarrow )$$ If $$f'(c)$$ exists, define $$\varphi$$ by $$\varphi (x) = \begin{cases} \frac{f(x)-f(c)}{x-c}&for \; x \neq c, x \in I\\ f'(c)&for \; x=c\end{cases}$$

$$\varphi$$ is continuous, as $$\lim_{x \rightarrow c} \varphi (x) = \varphi (c)$$ falls directly from the definition of the derivative. This definition also meets $$(1)$$ by multiplying both sides by $$x-c$$ and $$(2)$$ directly.

$$( \Leftarrow )$$ If $$\varphi$$ exists and is continuous, then $$ \varphi (c) = \lim_{x \rightarrow c} \varphi (x) = \lim_{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$$ exists. Therefore, the function is differentiable at c, and $$f'(c) = \varphi (c)$$.

QED