Continuity

What it is

A function f is continuous at the point a if three things hold: f(a) exists, the limit of f(x) as x approaches a exists, and the two are equal, that is, lim f(x) = f(a). Limits are explained on the limits page. A function is continuous on an interval if it is continuous at every point of the interval.

Examples of continuous functions:

• f(x) = x² is continuous at every point. At a = 3 the limit is 9 and f(3) = 9.
• Every polynomial, and sin x, cos x and eˣ.
• f(x) = |x| is continuous at 0 as well: the limit is 0 from both sides and f(0) = 0.
• f(x) = √x is continuous wherever it is defined, that is for x ≥ 0 (at 0 only the limit from the right counts, since the function has no values to the left).

Examples where continuity fails at a point:

• A hole: f(x) = (x² − 1)/(x − 1) has no value at x = 1. For x ≠ 1, f(x) = x + 1, so the limit at 1 is 2, but f(1) does not exist. Setting f(1) = 2 makes the function continuous; a break like this is called removable.
• The wrong value: f(x) = 1 for x ≠ 0 and f(0) = 0. The limit at 0 is 1 but the value is 0.
• A jump: f(x) = x for x < 1 and f(x) = x + 1 for x ≥ 1. From the left f(x) approaches 1, from the right 2, so the limit at 1 does not exist.
• The floor function ⌊x⌋ jumps by 1 at every integer. At x = 2 the value from the left is 1 and from the right 2, and f(2) = 2.

Why it matters

Continuity is the condition many theorems and methods rest on.

• Substitution. For a continuous function the limit is found by substituting: lim (x² + 1) as x approaches 3 is 3² + 1 = 10. This is why most limits met in a first course need no special trick, and L’Hôpital’s rule is needed only when substitution gives 0/0.
• The derivative. A function that is differentiable at a point is continuous there, so continuity is needed for the difference quotient in the definition of the derivative to have a limit. The converse does not hold, see below.
• The intermediate value theorem. If f is continuous on [a, b], f takes every value between f(a) and f(b). If f(a) and f(b) have opposite signs there is therefore a zero in between. Example: f(x) = x³ + 4x² − 10 has f(1) = −5 and f(2) = 14, so there is a root between 1 and 2 (near 1.365). The bisection method uses exactly this to halve the interval again and again.
• Reading graphs. In the graphing calculator the breaks show up as jumps, holes or vertical asymptotes.

Common misconceptions

• “Continuous means you can draw the graph without lifting the pencil.” As a picture over an interval it works, but it is not the definition. The function 1/x is continuous at every point where it is defined, that is for all x ≠ 0; its graph has two pieces because 0 is not in the domain.
• “Continuous means differentiable.” No. |x| is continuous at 0 but has no derivative there: the slope is −1 on the left and 1 on the right. The reverse does hold: differentiable at a point implies continuous there.
• “If the limit exists the function is continuous.” Not necessarily. For (x² − 1)/(x − 1) the limit 2 exists at x = 1, but the function has no value there. The limit must also equal the value.
• “Fractions with x in the denominator are discontinuous.” Such a function is continuous wherever it is defined. The break is at the point where the denominator is 0, which is not in the domain.

Common questions