1.12 Confirming Continuity over an Interval

Continuity at a point

A function f is continuous at x = c if three conditions hold: (1) f(c) exists, (2) lim(x→c) f(x) exists, and (3) lim(x→c) f(x) = f(c). If any condition fails, f is discontinuous at c. This builds directly on the limit definition from earlier lessons.

Continuity over an interval

A function is continuous over an open interval (a, b) if it's continuous at every point inside. For a closed interval [a, b], it must also be continuous at every interior point, plus right-continuous at a (lim x→a⁺ f(x) = f(a)) and left-continuous at b (lim x→b⁻ f(x) = f(b)).

Common function types

Polynomials, rational functions (where denominator ≠ 0), exponential, logarithmic, trigonometric, and radical functions are continuous on their natural domains. Combinations formed by addition, subtraction, multiplication, division (denominator ≠ 0), and composition of continuous functions are also continuous, letting you confirm continuity without computing limits directly.

Key terms

Three conditions for continuity at x = c

f(c) exists; lim(x→c) f(x) exists; lim(x→c) f(x) = f(c).

Continuity on an open interval (a,b)

f is continuous at every point strictly between a and b.

Continuity on a closed interval [a,b]

Continuous on (a,b), right-continuous at a, and left-continuous at b.

Right-continuous at a

lim(x→a⁺) f(x) = f(a).

Functions continuous on their domains

Polynomials, rational (denom ≠ 0), exponential, logarithmic, trig, and radical functions, plus sums, products, quotients, and compositions of continuous functions.

Try an AP-style question

Which function is continuous for all real numbers?

1. f(x) = 1/(x² − 1)
2. f(x) = tan x
3. f(x) = eˣ + x³
4. f(x) = √x