Carathéodory's Theorem (Analysis)

This proof is about Carathéodory's Theorem in the context of Analysis. For other uses, see Carathéodory's Theorem.

Theorem

Let $I \subseteq \R$.

Let $c \in I$ be an interior point of $I$.

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Let $f : I \to \R$ be a real function.

Then $f$ is differentiable at $c$ if and only if:

There exists a real function $\varphi : I \to \R$ that is continuous at $c$ and satisfies:

$(1): \quad \forall x \in I: \map f x - \map f c = \map \varphi x \paren {x - c}$

$(2): \quad \map \varphi c = \map {f'} c$

Proof

Necessary Condition

Suppose $f$ is differentiable at $c$.

Then by definition $\map {f'} c$ exists.

So we can define $\varphi$ by:

$\map \varphi x = \begin{cases}

\dfrac {\map f x - \map f c} {x - c} & : x \ne c, x \in I \\ \map {f'} c & : x = c \end{cases}$

Condition $(2)$, that $\varphi$ is continuous at $c$, is satisfied, since:

\(\ds \map \varphi c\)

\(=\)

\(\ds \map {f'} c\)

Definition of $\varphi$

\(\ds \)

\(=\)

\(\ds \lim_{x \mathop \to c} \frac {\map f x - \map f c} {x - c}\)

Definition of Derivative

\(\ds \)

\(=\)

\(\ds \lim_{x \mathop \to c} \map \varphi x\)

Definition of $\varphi$

Finally, condition $(1)$ is vacuous for $x = c$.

For $x \ne c$, it follows from the definition of $\varphi$ by dividing both sides of $(1)$ by $x - c$.

$\Box$

Sufficient Condition

Suppose a $\varphi$ as in the theorem statement exists.

Then for $x \ne c$, we have that:

$\map \varphi x = \dfrac {\map f x - \map f c} {x - c}$

Since $\varphi$ is continuous at $c$:

$\ds \map \varphi c = \lim_{x \mathop \to c} \map \varphi x = \lim_{x \mathop \to c} \frac {\map f x - \map f c} {x - c}$

That is, $f$ is differentiable at $c$, and $\map {f'} c = \map \varphi c$.

$\blacksquare$

Source of Name

This entry was named for Constantin Carathéodory.