Continuous

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

Formula

Formula

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.

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 1

Example

Solution

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

If x < 1,
f(x) = x2 + 1.

Piecewise Function

So the left-hand limit is
the limit of (x2 + 1) as x → 1-.

Find the limit value.

Put 1-
into (x2 + 1).

Limit of a Function

(1-)2 = 12

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 2

Example

Solution

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.