Differentiability & continuity

What you’ll learn

• Non-differentiability: Identify where a function fails to have a derivative using graphs or equations.
• Differentiability vs. continuity: Apply the principle that differentiability implies continuity, but continuity does not guarantee differentiability.

What is differentiability?

A function $f(x)$ is differentiable at $x=a$ if its derivative exists at that point.

There are four common cases where a function is not differentiable at a point $x=a$, meaning $f^{\prime }(a)$ does not exist:

1. Discontinuities

A function must be continuous to be differentiable. If there is a removable, jump, or infinite discontinuity at $x=a$, then the derivative does not exist at the point.

• Often seen in rational, piecewise, or logarithmic functions, which may have discontinuities and/or asymptotes
  e.g. $f(x)=\frac{1}{x}$

2. Corners

The function changes direction abruptly. The one-sided derivatives exist, but they are not equal.

• Often seen in absolute value or piecewise functions
  e.g. $f(x)=|x|$

3. Cusps

Cusps are like corners, but the slope becomes unbounded: one or both one-sided derivatives approach infinity, and they approach opposite infinities.

• Often seen in power functions with fractional exponents
  e.g. $f(x)=x^{2/3}$

4. Vertical tangents

The tangent line becomes vertical. The one-sided derivatives both approach infinity with the same sign.

• Often seen in radical functions
  e.g. $f(x)=x^{1/3}$

In the last three cases, the function is continuous, but the graph is not smooth at the non-differentiable point (for a visual, graph each in Desmos).

Remember this phrase:

$$\begin{gathered} \text{"If differentiable, then continuous."} \\ (D\Rightarrow C) \end{gathered}$$

Consequently, the contrapositive is always true:

If $f$ is not continuous at $x=a$, then it’s not differentiable at that point $(\text{not }C\Rightarrow \text{not }D)$.

However, the converse and inverse are not always true. A continuous function can still fail to be differentiable at certain points, e.g. $|x|$ at $x=0$.

Checking for differentiability

To check if $f(x)$ is differentiable at $x=a$,

1. Continuity: First check whether $f(x)$ is continuous by verifying the conditions stated in Section 1.6: Continuity. If it’s not continuous at $x=a$, then it’s not differentiable.

2. Differentiate: Find $f^{\prime }(x)$ and look for any $x$-values where $f^{\prime }(x)$ is undefined.

3. If necessary (mostly for piecewise functions): Check that the one-sided derivatives at $x=a$ both exist, are finite, and match. In other words, there must be a finite value $L$ such that

$$\underset{x\to a^{-}}{\lim }\frac{f(x)-f(a)}{x-a}=L=\underset{x\to a^{+}}{\lim }\frac{f(x)-f(a)}{x-a}$$

Example 1: From functions

For each of the following functions, find the $x$-value(s) at which it is non-differentiable.

a) $a(x)=\frac{1}{x+1}$

b) $b(x)=|x-1|$

c) $c(x)=x^{2/3}+1$

d) $d(x)=x^{1/3}$

Solutions

a) $a(x)=\frac{1}{x+1}$

b) $b(x)=|x-1|$

c) $c(x)=x^{2/3}+1$

d) $d(x)=x^{1/3}$

Example 2: From a graph

The graph of $f(x)$ shown below has a horizontal tangent at $x=b$ and a vertical tangent at $x=d$. At which of the values $a,b,c,$ and $d$ is $f(x)$ differentiable?

Solution

(spoiler)

Answer: $x=b$ only

A function is differentiable only where it is continuous and has a finite slope.

• $a$: Corner (different one-sided derivatives) $\Rightarrow$ not differentiable.
• $b$: Horizontal tangent ($f^{\prime }(b)=0$) $\Rightarrow$ differentiable.
• $c$: Jump discontinuity $\Rightarrow$ not differentiable.
• $d$: Vertical tangent ($f^{\prime }(d)$ doesn’t exist) $\Rightarrow$ not differentiable.

Example 3: Finding unknown variables

Let $f(x)$ be the piecewise function defined as:

$$f(x)=\{\begin{array}{ll}3x+2& \text{if}x<1\\ & \\ a{x}^2-bx& \text{if}x\ge 1\end{array}$$

Find the values of $a$ and $b$ such that $f(x)$ is differentiable for all values of $x$.

(spoiler)

Answer: $\boxed{a=-2,b=-7}$

For a piecewise function to be differentiable at $x=1$, it must be:

1. Continuous at $x=1$
2. Have equal left and right derivatives at $x=1$.

1. Continuity:

This requires

$$f(1)=\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }f(x)$$

• Function value:

$$f(1)=a(1{)}^2-b(1)=a-b$$

• Left-hand limit:

$$\underset{x\to 1^{-}}{\lim }(3x+2)=5$$

• Right-hand limit:

$$\underset{x\to 1^{+}}{\lim }(a{x}^2-bx)=a-b$$

So continuity requires

$$a-b=5$$

This provides one equation involving $a$ and $b$. To solve for both variables, we now use differentiability at $x=1$.

2. Differentiability:

Differentiate each piece (treating $a$ and $b$ as constants):

$$f^{\prime }(x)=\{\begin{array}{ll}3& \text{ if }x<1\\ & \\ 2ax-b& \text{ if }x>1\end{array}$$

For differentiability at $x=1$, the one-sided derivatives must match.

• Left-hand derivative: $\underset{x\to 1^{-}}{\lim }{f}^{\prime }(x)=3$

• Right-hand derivative: $\underset{x\to 1^{+}}{\lim }{f}^{\prime }(x)=2a-b$

Set equal:

$$3=2a-b$$

3. Solve the system:

$$\begin{array}{rl}a-b& =5\\ 2a-b& =3\end{array}\text{\hspace{1em}(1)(2)}$$

Solving the system of equations gives

$$\begin{gathered} a=-2 \\ b=-7 \end{gathered}$$

as the values for $f$ to be differentiable.

Graph check (optional):

Graphing

$$f(x)=\{\begin{array}{ll}3x+2& \text{ if }x<1\\ & \\ 2x^2+7x& \text{ if }x\ge 1\end{array}$$

in Desmos confirms the function is continuous and smooth at $x=1$.