# 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)## 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)At the point x = 1, there is a whole in the sub function g(x) = -x^{2} + 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:

## 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)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)