Differentiability

A function is said to be differentiable if a derivative exists for it over a given domain or a point. Many theorems only apply if a given function is differentiable over a domain, such as Rolle’s Theorem, Mean Value Theorem or L’Hôpital’s Rule.

Definition §

If $f$ is a real-valued function, then the derivative of $f$ at $x=a$ is defined by:

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$

which itself is derived from Derivation By First Principles.

$f$ is said to be differentiable at x = a if, and only if this limit exists.

Theorem - Differentiability and Continuity §

If $f$ is differentiable at $x=a$, then $f$ is also continuous at $x=a$:

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}⟹\underset{x\to a}{\lim }f(x)=f(a)$$

Using the contrapositive, discontinuity implies that a function is non-differentiable. I.e. if $f$ is not continuous over a point, then it is not differentiable either.

L’Hôpital’s Rule §