Types of Discontinuities

# Types of Discontinuities

We are now going to look at the three main types of discontinuities that can arise in a function. You should be able to distinguish between each type of discontinuity and acknowledge when a function $f$ may contain each type of discontinuity.

## Asymptotic Discontinuities

Asymptotic discontinuities arise when an asymptote exists. For example, let's look at the graph of the function $f(x) = \frac{1}{x + 1}$:

Notice that an asymptote exists at x = -1, because f(-1) = 1/0, which is indeterminate. As x → -1 from the left, f(x) → -∞, and as x → -1 from the right, f(x) → ∞. Hence it follows that we can say a function f(x) has a vertical asymptote if and only if:

(1)
\begin{align} \lim_{x \to a^-} = \pm \infty \quad \mathbf{and} \quad \lim_{x \to a^+} = \pm \infty \end{align}

## Point Discontinuities

Point discontinuities exist for piecewise functions where a specific value for x is defined differently than the rest of the piecewise function. For example, let's look at the following piecewise function:

(2)
\begin{align} f(x) = \left\{ \begin{array}{lr} -x^2 + 2 & , & x ≠ 2 \\ 1 & , & x = 2 \end{array} \right. \end{align}

At the point x = 1, there is a whole in the sub function g(x) = -x2 + 2, since when x = 2, f(2) = 1. f(2) ≠ g(2). In general, we say that a point discontinuity exists when for a function f(a) = b:

(3)
\begin{align} \lim_{x \to a^-} ≠ b \quad \mathbf{and} \quad \lim_{x \to a^+} ≠ b \end{align}

## Jump Discontinuities

Jump discontinuities are very similar to point discontinuities. Instead of a single point "jumping" from the normal curve, an entire portion or entire portions of the curve jump. For example, let's look at the following piecewise function:

(4)
\begin{align} f(x) = \left\{ \begin{array}{lr} x - 2 & , & x < 2 \\ x + 2 & , & x ≥ 2 \end{array} \right. \end{align}

Notice that a discontinuity occurs at x = 2, since the function f jumps from moving along x - 2, to moving along x + 2 starting at x = 2. In general we say that for a function f(x) such that f(a) = b, a jump discontinuity occurs when:

(5)
\begin{align} \lim_{x \to a^-} f(x) ≠ b \quad \mathbf{and} \quad \lim_{x \to a^+} f(x) = b \quad \mathbf{or,} \\ \lim_{x \to a^-} f(x) = b \quad \mathbf{and} \quad \lim_{x \to a^+} f(x) ≠ b \\ \end{align}