Continuity?

A function is continuous when you can draw its graph without ever lifting your pen — no jumps, no holes, no vertical breaks.

Simple Definition

Imagine tracing the graph of a function with your finger. If you can go from one end to the other without lifting your finger off the page, the function is continuous. The moment you have to jump, skip a point, or suddenly shoot off to infinity, you have a discontinuity.

Continuity is the mathematical formalization of "no surprises." At every point, the function arrives at exactly the value you'd predict by looking at the nearby values. The graph has no holes, no gaps, and no vertical walls.

This matters deeply for calculus: derivatives require continuity. You can't find the slope of a curve at a point where the curve breaks apart.

Intuition

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The pencil test

Draw the graph without lifting your pencil. If you can — it's continuous on that interval. This isn't just a metaphor; it's essentially what the mathematical definition captures. Continuity means the graph is one unbroken piece, everywhere you look.

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Real-world example

Temperature over time is continuous — it can't jump from 20°C to 35°C instantaneously. But the cost of a taxi ride (which changes in discrete steps per km) is discontinuous. Most physical phenomena are continuous; many economic ones are not.

The Three Conditions

A function $f$ is continuous at a point $x = a$ if and only if all three of the following hold:

$f(a)$ exists

The function is defined at $a$. No hole, no gap.

Limit exists

$\displaystyle\lim_{x \to a} f(x)$ exists — both sides agree.

They match

$\displaystyle\lim_{x \to a} f(x) = f(a)$

Formal Definition

$$f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)$$

Visual — Types of Discontinuity

Continuous — no breaks anywhere

Types of Discontinuity

Removable (Hole)

The limit exists but $f(a)$ is either undefined or doesn't match. Can be "fixed" by redefining the function at that one point. Example: $\frac{x^2-1}{x-1}$ at $x=1$.

Jump

Left-hand and right-hand limits both exist but are different. The graph literally jumps. Common in piecewise functions. Cannot be removed by redefining one point.

Infinite

The function shoots to $\pm\infty$ at the point — a vertical asymptote. Example: $\frac{1}{x}$ at $x=0$. The limit does not exist (it's $\pm\infty$).

Oscillatory

The function oscillates infinitely fast near the point and never settles. Example: $\sin(1/x)$ at $x=0$. The limit simply does not exist.

Key Theorems

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a,b]$ and $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c \in (a,b)$ such that $f(c) = k$.

In plain English: a continuous function can't skip values. If it's negative at one point and positive at another, it must cross zero somewhere in between.

Extreme Value Theorem (EVT)

If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains both a maximum and a minimum value on that interval.

Continuity guarantees the function doesn't escape to infinity — so it must reach its highest and lowest points.

Common Mistakes

✗ "The limit exists, so the function is continuous"

Not enough. The limit must equal the function's value at that point. A function with a hole at $x=2$ (limit = 5, but $f(2)$ undefined) is NOT continuous there, even though the limit exists.

✗ Confusing "continuous" with "differentiable"

Continuity is necessary but not sufficient for differentiability. $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$ — it has a sharp corner. Differentiable ⟹ continuous, but not vice versa.

✗ Assuming polynomials could be discontinuous

All polynomials, rational functions (away from their zeros in the denominator), exponentials, and trig functions are continuous on their entire domain. Discontinuities arise at domain restrictions, not from the algebra itself.

Related Concepts

Ready to test your knowledge?

Practice continuity and limits problems in the Calculus I quiz.

Go to Calculus I Quizzes →