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 §

Differentiability

If $f$ is a real-valued function, then $f$ is differentiable at some point $x=a$ if the limit below exists:

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

The limit is known as derivation by first principles. A function that is differentiable over its entire domain is said to be differentiable everywhere.

Infinitely Differentiable Functions §

Some functions can be differentiated an infinite number of times (over their domain). Mainly, this is because if the derivative of a function is $0$, we can still derivative it infinite times (getting $0$ each time). Such functions are infinitely differentiable:

• Polynomials
• Trigonometric Functions
• Exponential Functions
• Logarithms

Theorems §

T1: Differentiability implies continuity

$$f\text{ is differentiable at }a⟹f\text{ is continuous at }a$$

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

• 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.
• This theorem also means we can continuous functions that are non-differentiable. One such example is the Weierstrass function.