Continuity Examples

Locate where the following function is discontinuous, and classify each type of discontinuity.

The function isn't piecewise, so it doesn't jump (jump jump). The only other kinds of discontinuities we know how to find happen when the denominator equals zero. But lo, what to our wondering eyes does appear, but a denominator that cannot be zero. There are no x-values that make this function undefined. So, the function is always continuous.

Locate where the following function is discontinuous, and classify each type of discontinuity.

That denominator makes us wary. Let's factor it and see what shakes out.

x = -2 and x = -3 will both cause the function to be undefined. They are both discontinuities. However, we can cancel out (x + 2), so that's a hole, while x = -3 must be a vertical asymptote.

Is the following function continuous at x = 0?

At first glance, this thing actually looks like it could be continuous. So, in order to complete this problem, let's rely on our three parts to prove continuity. Grab something to drink and look over those one-sided limits.

False alarm. e⁰ = 1, not 0. So we can say with 100% certainty that this function is not continuous and has a jump discontinuity at x = 0.