# Continuous Functions

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

## Definition

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 *x* → *a*' 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

The left-hand limit of *f*(*x*) is

the limit of (*x*^{2} + 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

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

The left-hand limit of *f*(*x*) is

the limit of 2^{x} 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.