Lesson 22: Analyzing Continuous, Discontinuous, and Discrete Functions

New Concept

A continuous function is a function with a graph that has no gaps, jumps, or asymptotes.

Why it matters

Mastering function classification is key to advanced calculus and real-world modeling. Distinguishing between continuous and discrete systems is the first step toward analyzing everything from rocket trajectories to stock market jumps.

What’s next

Next, you'll learn to identify these function types from graphs and define their specific domains and ranges.

A continuous function is a function with a graph that has no gaps, jumps, or asymptotes. They are represented by smooth, unbroken curves or lines. A handy trick to remember is that you can draw the entire graph of a continuous function from start to finish without ever having to lift your pencil off the paper.

The graph of $f(x) = 2x - 1$ is a straight line, representing a smooth, unbroken path.
The graph of $g(x) = x^2$ is a parabola, which can be drawn entirely without lifting your pencil.
A function showing temperature change throughout a day is typically continuous, as it doesn't jump instantly.

Imagine you're drawing a rollercoaster track. If you can draw the whole thing without lifting your pen, it's continuous! It’s one smooth, unbroken ride with no sudden teleportation spots or mysterious gaps.

A discontinuous function is any function whose graph has gaps, jumps, or asymptotes, making it impossible to draw without lifting your pencil. These breaks can appear as holes, called points of discontinuity, or as vertical lines the graph approaches but never touches, known as asymptotes. It's essentially a graph with one or more interruptions in its path.

A function with a hole, like $f(x) = \frac{x^2 - 4}{x - 2}$, is discontinuous at $x = 2$.
A function with a vertical asymptote, such as $g(x) = \frac{1}{x}$, is discontinuous at $x = 0$.
A step function, which models costs like parking fees that jump at certain times, is discontinuous.

Think of a road with a broken bridge or a sudden sinkhole. You can't drive straight through! A discontinuous function's graph is just like that—it has breaks, forcing you to 'lift your pencil' to get from one part to the next.

A discrete function is a collection of separate, unconnected points. Instead of a smooth line or curve, its graph looks like a series of dots. This type of function is used when the input values are distinct and separate, such as counting objects, where you can't have fractional values. The domain and range consist of individual numbers.

The function representing the number of cars sold each day: $f = \{(1, 10), (2, 4), (3, 18), (4, 7)\}$.
A graph showing the number of students in each grade at a school would be discrete points.
The set of points $g = \{(1, 5), (2, 10), (3, 15)\}$ is a discrete function.

Picture crossing a river by hopping on stepping stones. You can only be on one stone at a time, not in the water between them. A discrete function is just like that—a set of separate points with nothing connecting them.