Continuity — Introduction

Take the function f(x) = x^2 and plot it. You get a smooth parabola — a single, unbroken curve that you can trace with your finger from left to right without ever lifting your hand off the screen. The graph has no gaps, no jumps, no sudden teleportations.

Now take this function:

This looks almost identical to f(x) = x^2, except at the single point x = 2. At that point, the parabola would give you 4, but g has been tampered with — someone reached in and moved the value at x = 2 up to 5. The graph has a hole at (2, 4) and a lonely dot floating at (2, 5).

If you tried to draw g without lifting your pen, you would be fine everywhere — until you reached x = 2. There, you would have to lift off the curve at the hole and stamp a dot at (2, 5). The graph is broken at that point.

The word for "not broken" is continuous. A function is continuous at a point if the graph passes through that point smoothly — no holes, no jumps, no explosions. The function f(x) = x^2 is continuous at x = 2. The function g(x) is not.

That is the intuition. But "no holes, no jumps, no explosions" is not a definition you can work with in mathematics. You need something precise enough to prove things with. The precise version is built from limits.

What continuity really means

Here is what goes wrong at x = 2 for the function g. As x approaches 2 — from either side — the values g(x) approach 4, because near x = 2 the function is just x^2. The limit is 4. But the actual value g(2) is 5. The limit and the value disagree.

That disagreement is the entire problem. For a function to be continuous at a point, the limit has to match the value. No disagreement. No surprise when you arrive.

Think of it like walking toward a door. As you get closer, you can see through the glass that the room inside looks a certain way. Continuity means that when you actually step through the door, the room is exactly what you saw through the glass. Discontinuity means the room suddenly looks different once you are inside — something changed at the threshold.

Three things can go wrong, and any one of them breaks continuity:

1. The function is not defined at the point. There is no room — the door leads to a wall.
2. The limit does not exist at the point. You cannot tell what the room looks like through the glass — different windows show different views.
3. The limit exists, but it does not equal the function's value. You can see through the glass just fine, but the room changes the instant you step inside.

Continuity at a point is the demand that none of these three things go wrong.

Condition 3 is the one that does the heavy lifting. It says: the value the function is approaching as you get close to a must be the same as the value the function actually takes at a. No surprises. No teleportation.

Notice that condition 3 implicitly requires conditions 1 and 2. If f(a) is not defined, you cannot write \lim_{x \to a} f(x) = f(a) — the right side is meaningless. If the limit does not exist, you cannot write the equality either. So some textbooks compress all three into the single statement \lim_{x \to a} f(x) = f(a). That is correct, but listing the three conditions separately is clearer when you are checking them one at a time.

Left and right continuity

Limits can be one-sided. A function can approach one value from the left and a different value from the right. When continuity depends on which side you approach from, you get one-sided continuity.

When would you care about this distinction? When the function behaves differently on the two sides of a point. The classic case is the greatest integer function f(x) = \lfloor x \rfloor — the function that takes any real number and rounds it down to the nearest integer.

Take x = 3. From the right, as x comes down from values like 3.1, 3.01, 3.001, the floor stays at 3. So \lim_{x \to 3^+} \lfloor x \rfloor = 3, which equals f(3) = 3. Right-continuous: yes.

From the left, as x approaches 3 through values like 2.9, 2.99, 2.999, the floor is 2 — always 2, right up until the instant x reaches 3. So \lim_{x \to 3^-} \lfloor x \rfloor = 2, which does not equal f(3) = 3. Left-continuous: no.

The floor function is right-continuous at every integer but not left-continuous. So it is discontinuous at every integer. Between integers — say at x = 3.7 — both one-sided limits equal 3, which is the function's value, so it is continuous there.

One-sided continuity becomes especially important when a function is defined on a closed interval [a, b]. At the left endpoint a, you can only approach from the right (there is nothing to the left), so you only need right-continuity there. At the right endpoint b, you only need left-continuity. This is exactly what happens next.

Continuity on an interval

A function that is continuous at a single point is useful, but the real power comes when a function is continuous across an entire stretch of the number line.

The boundary treatment is natural. At the endpoints, the interval only extends in one direction, so only the corresponding one-sided limit needs to match.

A function that is continuous on (-\infty, \infty) — the entire real line — is called continuous everywhere or simply a continuous function. Polynomials like x^2 - 3x + 7 are continuous everywhere. So are \sin x, \cos x, and e^x. These are the functions whose graphs have no breaks at all.

Checking continuity: a worked example

Time to apply the definition to an actual function and verify continuity step by step.

That example was intentionally simple. The three-condition check is trivial for polynomials — but the same framework applies to the hard cases. Here is one.

Examples of continuous and discontinuous functions

With the definition in hand, here is a catalogue of the standard cases.

Continuous everywhere

• Polynomials: x^3 - 4x + 7, 5x^2 + 1, and every polynomial of every degree. Limits of polynomials are computed by substitution, so the limit always equals the value.

• Sine and cosine: \sin x and \cos x are continuous on the entire real line. Their graphs are smooth waves with no breaks.

• Exponential functions: e^x, 2^x, 10^x are continuous on (-\infty, \infty). They grow (or decay) smoothly without interruption.

Continuous on their domain, discontinuous outside

• f(x) = 1/x: Continuous at every x \neq 0. At x = 0 the function is not defined (condition 1 fails), and the limit does not exist either — f(x) shoots to +\infty from the right and -\infty from the left.

• f(x) = \sqrt{x}: Continuous on [0, \infty). At x = 0 it is right-continuous. For negative x, the function is not defined in the reals, so the question of continuity does not arise.

• f(x) = \tan x: Continuous everywhere except at x = \pm\pi/2, \pm 3\pi/2, \ldots — the points where \cos x = 0 and the function shoots to infinity.

Discontinuous at specific points

• The floor function \lfloor x \rfloor: Discontinuous at every integer. Between integers it is continuous — just a constant, after all.

• The sign function \text{sgn}(x): Equals -1 for x < 0, 0 for x = 0, and 1 for x > 0. Discontinuous at x = 0: the left limit is -1, the right limit is +1, and the function value is 0. None of the three match each other.

• Tampered functions: The function g(x) from the start of this article — x^2 everywhere except g(2) = 5 — is discontinuous at x = 2 because the limit is 4 but the value is 5. This type of break is called a removable discontinuity, because you can fix it by redefining the function at that single point. More on this in the article on types of discontinuity.

Common confusions

• "Continuous means smooth." Not the same thing. A function with a sharp corner — like |x| at the origin — is continuous (the graph is unbroken) but not smooth (the graph has a kink). Continuity is about the graph having no gaps; smoothness is about the graph having no corners. Smooth is a stronger condition.

• "If a function is defined at a point, it is continuous there." No. The function g(x) above is defined at x = 2 (its value is 5), but it is not continuous there because the limit (4) disagrees with the value (5). Being defined is only one of the three conditions.

• "Continuity means I can draw the graph without lifting my pen." This is excellent intuition and it is essentially correct for the functions you will encounter in school mathematics. But be careful: it is a visual heuristic, not a definition. Some pathological functions (like the Dirichlet function, which equals 1 at rationals and 0 at irrationals) are discontinuous everywhere — and you certainly cannot draw that graph — but the pen metaphor does not help you prove that. The three-condition definition does.

• "The limit existing is enough for continuity." The limit existing is only condition 2. You also need the function to be defined there (condition 1) and the limit to equal the value (condition 3). The function h(x) = (x^2 - 4)/(x - 2), which simplifies to x + 2 for x \neq 2, has a limit of 4 at x = 2 — but h(2) is not defined (division by zero), so the function is not continuous at x = 2.

Going deeper

The epsilon-delta definition

The definition above used limits, and limits themselves have a rigorous formulation in terms of \varepsilon (epsilon) and \delta (delta). When you unwind the limit in the continuity definition, you get:

f is continuous at x = a if for every \varepsilon > 0, there exists a \delta > 0 such that

Read it carefully. It says: no matter how small a tolerance \varepsilon you set for the output, there is a window of width 2\delta around a in the input such that every x in that window produces an f(x) within \varepsilon of f(a).

This is the same three-condition check, compressed into one formal sentence. The existence of f(a) is implicit (you are writing f(a), so it had better exist). The existence of the limit is guaranteed by the \delta that works for every \varepsilon. And the equality of the limit and the value is built into the |f(x) - f(a)| < \varepsilon — the function values near a cluster around f(a) itself, not around some other number.

Verifying epsilon-delta for f(x) = 3x + 1 at x = 2

Take \varepsilon > 0. You need |f(x) - f(2)| < \varepsilon, i.e., |3x + 1 - 7| < \varepsilon, i.e., |3x - 6| < \varepsilon, i.e., 3|x - 2| < \varepsilon. This holds whenever |x - 2| < \varepsilon/3. So choose \delta = \varepsilon/3. For every \varepsilon, this \delta works, and the definition is satisfied. The function is continuous at x = 2.

This is the skeleton of every epsilon-delta proof of continuity. You start from the inequality |f(x) - f(a)| < \varepsilon, manipulate it until you get |x - a| < \text{something}, and then declare that "something" to be your \delta.

Why continuity matters: the big theorems

Continuity is not just a hygiene check. It is the hypothesis of some of the most important theorems in calculus.

The Intermediate Value Theorem (IVT). If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then there is some c in (a, b) where f(c) = 0. In plain language: a continuous function that starts positive and ends negative (or vice versa) must cross zero somewhere in between. It cannot "jump over" zero without passing through it — because a continuous function has no jumps.

This is how you prove that equations have solutions without actually solving them. For instance, x^3 - x - 1 = 0 has a root between 1 and 2, because f(1) = -1 < 0 and f(2) = 5 > 0, and f is a polynomial (hence continuous).

The Extreme Value Theorem. If f is continuous on a closed interval [a, b], then f attains both a maximum and a minimum value somewhere in that interval. A continuous function on a closed interval cannot escape to infinity or oscillate without settling — it is bounded and achieves its bounds.

Both theorems fail the moment you drop continuity. A function with a single jump discontinuity can cross from positive to negative without ever being zero (it jumps over zero). A function with an infinite discontinuity can blow up without having a maximum. Continuity is not decoration — it is the structural condition that makes calculus work.

Where this leads next

You now know what continuity means at a point, on an interval, and from one side. The next articles build on this.

• Types of Discontinuity — the four ways continuity can fail, each with a different character: removable, jump, infinite, and oscillatory.
• Properties of Continuous Functions — the algebra of continuous functions (sums, products, quotients, composites), the continuity of standard functions, and piecewise continuity.
• Derivative — the derivative requires continuity as a prerequisite. A function that is differentiable at a point is always continuous there, but not the other way around.
• Intermediate Value Theorem — the full statement and proof of the IVT, with applications to root-finding.
• Algebra of Limits — the limit laws that underpin every continuity check you have seen in this article.