Continuous Functions

Continuous Functions

How to solve continuous functions problems: definition, examples, and their solutions.

Definition

If 'the limit of f(x) as x goes to a' and 'f(a)' are equal, then f(x) is continuous at x = a.

A continuous function is a function
that is not disconnected.

In this graph,
f(x) is not disconnected at x = a.

So f(x) is continuous at x = a.

This means
'the limit of f(x) as xa' and f(a)
are equal.

If the left-hand piecewise function of f(x)
and the right-hand piecewise function of f(x)
are given differently,
set an equation like this:

the left-hand limit of f(x),
the right-hand limit of f(x),
and f(a)
are all equal.

One-sided limits

Example 1

Determine whether f(x) is continuous at x = 1 or not. f(x) = x^2 + 1 (x < 1), -x + 3 (x >= 1).

The left-hand limit of f(x) is
the limit of (x2 + 1) as x → 1-,
which is equal to 2.

Piecewise functions

The right-hand limit of f(x) is
the limit of (-x + 3) as x → 1+,
which is equal to 2.

f(1) is 2.

The left-hand limit (2), the right-hand limit (2),
and the function value (2) are all equal.

So f(x) is continuous at x = 1.

See the graph of f(x).

You can see that
the function is not disconnected at x = 1.

So f(x) is continuous at x = 1.

Example 2

Determine whether f(x) is continuous at x = 3 or not. f(x) = (x + 1)(x - 3)/(x - 3) (x is not equal to 3), 2 (x = 3).

The limit of f(x) as x → 3 is
the limit of (x + 1)(x - 3)/(x - 3) as x → 3,
which is equal to 4.

Piecewise functions

f(3) is 2.

The limit (4) and the function value (2) are not equal.

So f(x) is not continuous at x = 3.

See the graph of f(x).

You can see that
the function is disconnected at x = 3.

So f(x) is not continuous at x = 3.

Example 3

Find the values of a and b that make f(x) continuous f(x) = 2^x (x > 0), a (x = 0), cos x + b (x > 0).

The left-hand limit of f(x) is
the limit of 2x as x → 0-,
which is equal to 1.

Piecewise functions

f(0) is a.

The right-hand limit of f(x) is
the limit of (cos x + b) as x → 0+,
which is equal to 1 + b.

It says f(x) should be continuous.

So the left-hand limit (1), the function value (a),
and the right-hand limit (1 + b) should be all equal.

So 1 = a = 1 + b.

So a = 1 and b = 0.