Continuity

Level 1 - Calculus topic page in Foundations And Limits.

Principle

A function is continuous at a point when its graph has no break there. Algebraically, the function value must exist, the limit must exist, and the two must agree.

Notation

\(f(a)\)

value of f at x=a

\(\lim_{x\to a}f(x)\)

limit of f near a

\(a\)

point where continuity is tested

The Core Method

Use the three-part continuity test.

Continuity at a

\[f\text{ is continuous at }a\Longleftrightarrow \lim_{x\to a}f(x)=f(a)\]

This statement includes the requirement that both sides exist. For piecewise functions, compare the left-hand limit, right-hand limit, and defined value.

Worked Cases

Question

Let \(f(x)=x^2\) for \(x<1\), and \(f(x)=kx+1\) for \(x\ge1\). Find \(k\) so that \(f\) is continuous at \(1\).

Answer

The left-hand limit is \(1^2=1\). The value from the right-hand formula is \(f(1)=k+1\). Continuity requires \(k+1=1\), so \(k=0\).

Examples

Question

Is \(f(x)=\frac{x^2-1}{x-1}\) continuous at \(x=1\)?

Answer

The formula is not defined at \(x=1\), because the denominator is zero. Although the simplified expression is \(x+1\) for \(x\ne1\), the value \(f(1)\) does not exist. Therefore \(f\) is not continuous at \(1\).

Limit Laws

Infinite Limits