Carathéodory's Theorem (Analysis)

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This proof is about Carathéodory's Theorem in analysis. For other uses, see Carathéodory's Theorem.


Theorem

Let $I \subseteq \R$, and let $c \in I$ be an interior point of $I$.



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

Then $f$ is differentiable at $c$ iff:

There exists a real function $\varphi : I \to \R$ that is continuous at $c$ and satisfies:
$(1): \quad \forall x \in I: f \left({x}\right) - f \left({c}\right) = \varphi \left({x}\right) \left({x - c}\right)$
$(2): \quad \varphi \left({c}\right) = f' \left({c}\right)$


Proof

Necessary Condition

Suppose $f$ is differentiable at $c$.

Then by definition $f' \left({c}\right)$ exists.

So we can define $\varphi$ by:

$\varphi \left({x}\right) = \begin{cases} \dfrac {f \left({x}\right) - f \left({c}\right)} {x - c} & : x \ne c, x \in I \\ f' \left({c}\right) & : x = c \end{cases}$

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

\(\displaystyle \varphi \left({c}\right)\) \(=\) \(\displaystyle f' \left({c}\right)\)          Definition of $\varphi$          
\(\displaystyle \) \(=\) \(\displaystyle \lim_{x \to c} \frac {f \left({x}\right) - f \left({c}\right)} {x - c}\)          Definition of derivative          
\(\displaystyle \) \(=\) \(\displaystyle \lim_{x \to c} \varphi \left({x}\right)\)          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:

$\varphi \left({x}\right) = \dfrac {f \left({x}\right) - f \left({c}\right)} {x - c}$

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

$\displaystyle \varphi \left({c}\right) = \lim_{x \to c} \varphi \left({x}\right) = \lim_{x \to c} \frac {f \left({x}\right) - f \left({c}\right)} {x - c}$


That is, $f$ is differentiable at $c$, and $f' \left({c}\right) = \varphi \left({c}\right)$.

$\blacksquare$


Source of Name

This entry was named for Constantin Carathéodory.