# Subsection6.2.1Differentiability

A function is differentiable at points where the derivative is defined. Alternatively, because the derivative at a point represents the slope of the tangent line, we say the function is differentiable at a point wherever the function has a well-defined tangent line.

##### Definition6.2.1Differentiability

A function $f$ is differentiable at $a$ if $f'(a)$ exists, or more precisely the limit \begin{equation*}\lim_{h \to 0} \frac{f(a+h)-f(a)}{h} = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\end{equation*} exists.

A function is not differentiable if the limit does not exist. There are several reasons this might occur. The first reason is if the function is not continuous.

##### Proof

Another way that a function might not have be differentiable is where it is continuous but has a corner. This means that the slope at the point looks different from either of the two sides. Mathematically, if we computed the one-sided limits for the formula of the derivative, we would get two different values.

##### Example6.2.3

Consider the piecewise function defined by \begin{equation*}f(x) = \begin{cases} x^2, & x \le 1, \\ x, & x \gt 1.\end{cases}\end{equation*} Determine if $f$ is differentiable at $x=1$.

Solution
##### Example6.2.4

Consider the piecewise function defined by \begin{equation*}f(x) = \begin{cases} x^2-3x+8 & x \lt 2, \\ 5x-x^2, & x \ge 2.\end{cases}\end{equation*} Determine if $f$ is differentiable at $x=2$.

Solution

# Subsection6.2.2Consequences of Differentiability

There are a number of important consequences of a function being differentiable. These consequences are stated as mathematical theorems that you will need to know by name. We begin by introducing terminology about local extreme values.

##### Definition6.2.5Local Maximum and Minimum

A function $f$ has a local maximum at a point $x=a$ if $f(a) \ge f(x)$ for all $x$ in a neighborhood of $a$. It has a local minimum at $x=a$ if $f(a) \le f(x)$ for all $x$ in a neighborhood of $a$.

The first theorem is about the slope at a local extreme. It guarantees that a local extreme can only occur where the function either is not differentiable or has a horizontal tangent line.

The second theorem combines the Extreme Value Theorem with Fermat's Theorem. If a function is continuous on a closed interval $[a,b]$, then it must achieve both a maximum and a minimum value. If that function has $f(a)=f(b)$, then one of the extreme values must occur inside the interval at some point $c \in (a,b)$. If the function is also differentiable, then we must have $f'(c)=0$. This result is named Rolle's theorem.

The consequence of Rolle's theorem is that if a function starts and ends at the same value over an interval, it must turn around somewhere. For a differentiable function, the slope at that point must be $f'(c)=0$.

The third theorem about differentiability applies Rolle's theorem to say something about the average rate of change. Recall that the average rate of change, \begin{equation*}\left.\frac{\Delta f}{\Delta x}\right|_{[a,b]} = \frac{f(b)-f(a)}{b-a},\end{equation*} is the slope of the line, called a secant line, that joins the points $(a,f(a))$ and $(b,f(b))$. The Mean Value Theorem guarantees that a continuous and differentiable function will have some point at which the tangent line has the same slope as the secant line over the interval.