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

Graphical or geometric meaning

The original rational rule traces the line $y=x+2$ with a hole at $(2,4)$. Setting $k=4$ fills exactly that hole. A jump has mismatched one-sided limits; an infinite discontinuity has unbounded output near the point.

Common mistakes and why they fail

Verification and reasonableness checks

• Evaluate all three conditions separately at a suspicious point.
• Use known continuity rules only where component functions are defined.
• Compare a symbolic limit with a graph or numerical table.

Choosing the right continuity test

Continuity is checked at a specified point relative to the domain. At an interior point $a$, verify that $f(a)$ exists, $\lim_{x\to a}f(x)$ exists, and they agree. At a domain endpoint, use the available one-sided limit. For a piecewise formula, the only nonautomatic checks are usually the joining values. A removable discontinuity differs from a jump or vertical asymptote because its two-sided limit exists even when the displayed value is missing or wrong. Redefining one point can fix the removable case but not a jump. A graph may suggest the type; the limit calculation supplies the proof. State the domain before deciding which approaches to $a$ are meaningful.

Practice

1. Is $1/x$ continuous at $0$?
2. Is $|x|$ continuous at $0$?
3. Choose $c$ so $f(x)=x+1$ for $x<2$ and $f(x)=c$ at $x=2$ is left-continuous.

Connections and next steps

• Limits
• One-Sided Limits
• Intermediate Value Theorem