Skill: Identify discontinuities

A function is discontinuous at $x=a$ if at least one of these occurs:

1. $f(a)$ is undefined.
2. The two-sided limit $\displaystyle \lim_{x\to a}f(x)$ does not exist.
3. The limit exists, but it is not equal to $f(a)$.

To identify discontinuities, check:

• where a denominator is zero,
• where a piecewise formula changes,
• where there is a missing point or specially defined value.

Worked example

Consider

$$f(x)= \begin{cases} \dfrac{x^2-1}{x-1}, & x<2,\ x\ne 1,\\[6pt] 5, & x=1,\\ 4, & x=2,\\[4pt] \dfrac{1}{x-3}, & x>2. \end{cases}$$

We will check the possible problem points $x=1$, $x=2$, and $x=3$.

Step 1: Check $x=1$

For values near $1$, use

$$\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1, \qquad x\ne 1.$$

Therefore,

$$\lim_{x\to 1}f(x) = \lim_{x\to 1}(x+1) = 2.$$

However, the function is defined separately as

$$f(1)=5.$$

Since the limiting value $2$ is different from the function value $5$, there is a removable discontinuity, or hole, at $x=1$.

Step 2: Check $x=2$

Find the one-sided limits.

From the left, use the first rule:

$$\lim_{x\to 2^-}f(x) = \lim_{x\to 2^-}(x+1) = 3.$$

From the right, use the last rule:

$$\lim_{x\to 2^+}f(x) = \lim_{x\to 2^+}\frac{1}{x-3} = -1.$$

The one-sided limits are different:

$$3\ne -1.$$

Thus, the two-sided limit does not exist. Even though $f(2)=4$ is defined, the function has a jump discontinuity at $x=2$.

Step 3: Check $x=3$

Near $x=3$, the function is

$$f(x)=\frac{1}{x-3}.$$

As $x$ approaches $3$ from the left,

$$\lim_{x\to 3^-}\frac{1}{x-3}=-\infty,$$

and from the right,

$$\lim_{x\to 3^+}\frac{1}{x-3}=+\infty.$$

Also, $f(3)$ is undefined because the denominator is zero. Therefore, there is an infinite discontinuity, associated with the vertical asymptote

$$x=3.$$

Conclusion

The function has:

• a removable discontinuity at $x=1$,
• a jump discontinuity at $x=2$,
• an infinite discontinuity at $x=3$.

The main test is to compare the function value with the limit, and when necessary, compare the left-hand and right-hand limits.