# Continuous

How to show that the given function is continuous: formula, 2 examples, and their solutions.

## Definition

If a function is continuous at a point,

then the function is not disconnected at that point.

So, if the limit value and the function value

are equal at x = a,

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

If a function is continuous at every point,

then the function is a continuous function.

## Formula

The limit value exists

if the left-hand limit and the right-hand limit

are equal.

One-Sided Limits

So, if the left-hand limit, the right-hand limit,

and the function value are all equal,

then f(x) is continuous.

## Example

Find the left-hand limit of f(x) at x = 1.

If x < 1,

f(x) = x^{2} + 1.

Piecewise Function: Graph

So the left-hand limit is

the limit of (x^{2} + 1) as x → 1^{-}.

Find the limit value.

Put 1^{-}

into (x^{2} + 1).

Limit of a Function

(1^{-})^{2} = 1^{2}

1 + 1 = 2

So the left-hand limit is 2.

Find the right-hand limit of f(x) at x = 1.

If x ≥ 1,

f(x) = -x + 3.

So the right-hand limit is

the limit of (-x + 3) as x → 1^{+}.

Find the limit value.

Put 1^{+}

into (-x + 3).

-(1^{+}) = -1

-1 + 3 = 2

So the right-hand limit is 2.

Find the function value f(1).

If x ≥ 1,

f(x) = -x + 3.

So f(1) = -1 + 3 = 2.

The left-hand limit is 2.

The right-hand limit is 2.

And the function value f(1) is 2.

So the left-hand limit, the right-hand limit,

and the function value are all equal

at x = 1.

Then y = f(x) is continuous at x = 1.

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

## Example

Find the limit of f(x) at x = 3.

If x ≠ 3,

f(x) = (x + 1)(x - 3)/(x - 3).

So the limit value is

the limit of (x + 1)(x - 3)/(x - 3) as x → 3.

Cancel (x - 3) factors.

Find the limit value.

Put 3

into (x + 1).

Then 3 + 1 = 4.

So the limit value is 4.

Find the function value f(3).

If x = 3,

f(x) = 2.

So f(3) = 2.

The limit value is 4.

And the function value f(3) is 2.

So the limit value and the function value

are not equal

at x = 3.

Then y = f(x) is not continuous at x = 3.

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