Continuity

The central idea

A function $f$ is continuous at $a$ when three conditions hold: $f(a)$ is defined, $\lim_{x\to a}f(x)$ exists, and that limit equals $f(a)$. It is continuous on an open interval if it is continuous at every point, and on a closed interval $[a,b]$ using the appropriate one-sided conditions at the endpoints.

Definitions, hypotheses, and notation

Continuity behaves well under sums, products, compositions, and quotients whose denominators do not vanish. These closure rules justify direct substitution for polynomials, exponential functions, and many composites, but only on their domains. A rational expression is not continuous at a denominator zero simply because an algebraically canceled formula extends there; the extension must be defined separately.

Different discontinuities require different repairs. A removable discontinuity has a finite limit and can be filled by redefining one value. A jump has unequal finite one-sided limits, so changing only the join value cannot make it continuous. An infinite discontinuity has unbounded nearby values. Naming the type identifies which of the three defining conditions fails and whether a pointwise repair is possible.

Conceptual meaning

Continuity means nearby inputs produce nearby outputs. Graphically, there is no hole, jump, or vertical blow-up at the point. It does not mean a graph has no corner: $|x|$ is continuous at zero even though it is not differentiable there.

A dependable method and decision rule

1. Check that the point belongs to the function's domain.
2. Compute the left- and right-hand limits; they must agree for an interior point.
3. Compare the common limit with the actual function value.
4. For piecewise functions, impose equality of the formulas at the join.
5. State which continuity condition fails if the function is discontinuous.

Fully worked example

Common mistakes and why they fail