Discontinuous functions are functions that are not continuous at one or more inputs, usually because of a hole, jump, or break in the graph. In Calculus I, you check them when studying limits and continuity.
In Calculus I, a discontinuous function is a function that fails the continuity test at at least one point. That means the graph has a hole, jump, or break, or the function is not defined at a value where you would expect it to behave smoothly.
The big idea is that continuity asks for three things at a point: the function has to exist there, the limit has to exist there, and the limit has to match the actual function value. If any one of those fails, the function is discontinuous at that input. So discontinuity is not just about a weird-looking graph, it is about a precise failure in how the function behaves near a number.
A common case is a removable discontinuity, which looks like a hole. For example, if a rational expression simplifies after factoring, the simplified version may show the graph should have one missing point because the original formula was undefined there. You can sometimes fix that by defining the function value at that input to match the limit.
Another common case is a jump discontinuity, where the left-hand behavior and right-hand behavior land at different values. This shows up in piecewise functions, especially when two formulas meet but do not line up. The graph stays defined on both sides, but it does not connect at that input.
You will also see discontinuities from vertical asymptotes, absolute-value style corners are not discontinuities, but division by zero and piecewise mismatches often are. The point is to read the graph or formula carefully and ask, “What happens as x gets close to this number from each side?” That limit-based thinking is what separates a vague break in the picture from a real Calculus I classification.
Discontinuous functions show up right when Calculus I starts connecting algebra to limits. If you cannot tell whether a function is continuous, you cannot safely evaluate a limit by direct substitution, and you will miss why some functions behave nicely while others do not.
This term also sets up derivative work. A function cannot be differentiable at a point where it is discontinuous, so spotting a break early saves you from trying to take slopes where the graph does not behave smoothly. That matters later in curve sketching, optimization, and related rates, where you need to know where a function is defined and where it might fail.
Discontinuities also make piecewise functions much easier to read. When you are given a graph or an equation split into cases, the first question is often whether the pieces meet cleanly or create a hole or jump. That tells you how to classify the function and what happens to the limit at the boundary point.
In problem solving, discontinuity is often the reason you factor, simplify, or compare one-sided limits. A hole might be fixable, a jump is not, and a vertical asymptote changes the whole picture of the graph. Being able to name the type of discontinuity gives you a shortcut for describing the function’s behavior instead of just saying, “it looks broken.”
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view galleryContinuity
Continuity is the opposite idea you check against. A function is continuous at a point only if it is defined there, the limit exists there, and the limit equals the function value. Discontinuous functions fail at least one of those conditions, so continuity gives you the exact checklist for deciding whether a graph is smooth at a point.
Jump Discontinuity
A jump discontinuity is one specific kind of discontinuity where the left-hand and right-hand limits both exist, but they are different. That means the graph has two finite sides that do not meet. In Calculus I, this often shows up in piecewise functions with a boundary that does not line up.
Removable Discontinuity
A removable discontinuity is the type that looks like a hole. The limit exists, but the function is missing the matching value, or the formula is undefined at that input. This is the discontinuity you can sometimes fix by redefining the function at the missing point.
Factoring
Factoring often helps you detect removable discontinuities in rational functions. When a numerator and denominator share a factor, you can simplify the expression and see whether a canceled value created a hole. In limit problems, that algebra move is often the fastest way to understand what the graph is doing near the bad input.
A quiz problem might give you a graph or formula and ask you to identify where the function is discontinuous, then name the type. You may also be asked to use factoring or one-sided limits to decide whether a hole is removable or whether a jump is present. On problem sets, you often justify your answer with the limit from the left, the limit from the right, and the function value at the point. If the function is piecewise, the boundary point is where you check most carefully. A correct answer is usually not just the location of the break, but the reason the continuity test fails.
These are opposites, and they are easy to mix up because both describe how a function behaves at a point. Continuity means the function passes the three-part check at that input, while discontinuity means at least one part fails. If you remember the continuity test, you can identify discontinuity by spotting exactly which condition breaks.
A discontinuous function is not continuous at one or more inputs, so its graph has a hole, jump, break, or other failure at that point.
In Calculus I, discontinuity is checked with the continuity test: the function must be defined there, the limit must exist, and the limit must equal the function value.
A removable discontinuity is a hole that may be fixed by redefining the function at the missing input.
A jump discontinuity happens when the left-hand and right-hand limits land at different values.
If a function is discontinuous at a point, it cannot be differentiable there.
A discontinuous function is a function that fails continuity at one or more inputs. In Calculus I, that usually means the graph has a hole, jump, or break, or the function is undefined at a point you need to check. You decide that by using the continuity test and, when needed, one-sided limits.
Check the point where the graph or formula looks suspicious. Ask whether the function is defined there, whether the limit exists from both sides, and whether the limit matches the actual function value. If any part fails, the function is discontinuous at that input.
A removable discontinuity is a hole, which means the limit exists but the function value is missing or mismatched. A jump discontinuity happens when the left and right limits are different, so the graph cannot be fixed by filling in one missing point. That makes jump discontinuities more than just a missing value.
Not at the point where it is discontinuous. Differentiability requires continuity first, so a break, hole, or jump blocks the derivative there. The function may still be differentiable at other points where it is continuous and smooth.