A discontinuous function is a function whose graph has a break, jump, or hole at one or more input values. In Intermediate Algebra, you spot these gaps by looking at the graph, the equation, or both.
A discontinuous function is a function in Intermediate Algebra whose graph is not connected everywhere in its domain. That means there is at least one x-value where the function does not flow smoothly from left to right. Instead, you may see a hole, a jump, or a break in the graph.
The easiest way to picture this is to imagine tracing the graph with your pencil. If you have to lift your pencil because the graph skips a point or suddenly changes height, the function is discontinuous there. The graph might still be a function, since the vertical line test can still pass, but it is not continuous at every point.
A common example is a rational function with a denominator that becomes 0 at some x-value. That x-value is not allowed in the domain, so the graph has a gap or a vertical asymptote there. Another example is a piecewise function that uses one rule on one interval and a different rule on another interval. If the pieces do not meet at the same y-value, the graph has a jump discontinuity.
Not every discontinuity looks the same. A removable discontinuity is a hole, often caused by a factor that cancels in an expression. A jump discontinuity happens when the graph approaches two different heights from the left and right. In Intermediate Algebra, you usually do not need formal limit notation, but you do need to tell these apart by looking at the graph or by checking what happens around the missing point.
The big idea is that discontinuous functions still follow the rules of functions, but they do not stay smooth across all x-values. When you graph them, you are looking for where the pattern breaks and why it breaks. That makes discontinuity a useful clue for domain, graph shape, and later work with rational and piecewise functions.
Discontinuous functions show up any time you work with graphs that have restrictions, holes, or separate pieces. In Intermediate Algebra, that usually means rational expressions, piecewise definitions, or graphs that do not behave like a single smooth curve. If you can spot the discontinuity, you can figure out the domain more accurately and avoid using input values that make the expression undefined.
This concept also sharpens your graph-reading skills. A hole on the graph tells you one thing, while a jump tells you something different. Those details matter when you are asked to describe behavior, match an equation to a graph, or explain why a certain x-value is excluded.
Discontinuities also connect directly to later algebra topics. When you simplify a rational expression, you may cancel a factor and uncover a removable discontinuity. When you graph piecewise functions, you have to check whether the pieces meet or leave a gap. That kind of careful checking is a repeated skill in algebra because one small break can change the whole graph.
Keep studying Intermediate Algebra Unit 3
Visual cheatsheet
view galleryContinuous Function
A continuous function is the opposite idea: its graph stays connected with no breaks, holes, or jumps in the interval you are looking at. Comparing the two helps you decide whether a graph flows smoothly or has a place where the rule fails. If a function is not continuous at a point, it is discontinuous there.
Jump Discontinuity
A jump discontinuity is one specific kind of discontinuity. The graph approaches one y-value from the left and a different y-value from the right, so the function seems to 'jump' to a new height. In Intermediate Algebra, this often shows up in piecewise functions where two rules do not meet at the same point.
Removable Discontinuity
A removable discontinuity is the classic hole in a graph. It often appears when a factor cancels in a rational expression, leaving the function undefined at one x-value even though the simplified expression looks fine. You can think of it as a missing point that could be filled in if the rule were written differently.
Vertical Line Test
The vertical line test checks whether a graph is a function, not whether it is continuous. A graph can pass the vertical line test and still have a discontinuity. That is why a function can have a hole or a jump and still count as a function.
A quiz or test problem might show you a graph and ask you to identify where the function is discontinuous, or to name the type of discontinuity. You may also be given an equation and asked to find the x-value that makes the denominator 0, then decide whether that creates a hole or another kind of break. In graphing problems, look for open circles, sudden jumps, and places where the graph is not defined. If the function is piecewise, check whether the pieces meet at the boundary point. The answer usually comes from noticing the exact x-value where the graph stops behaving smoothly, not from guessing based on the overall shape.
These are easy to mix up because both describe how a graph behaves across x-values. A continuous function has no breaks in the interval, while a discontinuous function has at least one break, hole, or jump. If you are checking a graph, ask whether you can trace it without lifting your pencil at that point.
A discontinuous function has at least one point where the graph breaks, jumps, or has a hole.
In Intermediate Algebra, discontinuity often shows up in rational expressions and piecewise functions.
A function can still pass the vertical line test and be discontinuous at one or more x-values.
Removable discontinuities look like holes, while jump discontinuities show different left-hand and right-hand behavior.
Finding a discontinuity usually means checking where the expression is undefined or where the graph stops matching smoothly.
It is a function whose graph does not stay connected at every x-value in its domain. You might see a hole, a jump, or a break in the graph. In algebra, this usually comes up when an expression is undefined at a certain input or when a piecewise graph changes rule suddenly.
Look for open circles, gaps, vertical asymptotes, or places where the graph jumps to a different height. If you cannot trace the graph smoothly through a point, that point is discontinuous. The graph can still be a function, so use the vertical line test separately.
A hole is a removable discontinuity, which means the graph is missing one point but the left and right sides may still meet at the same height. A jump discontinuity happens when the left side and right side approach different y-values. The graph does not just miss a point, it changes height abruptly.
Yes. Discontinuous only means the graph has a break, not that it fails the definition of a function. As long as each x-value has only one output, it can still be a function. That is why a graph can pass the vertical line test and still have a discontinuity.