Ximpledu

Continuous Function

See how to find out if a function is a continuous function.
2 examples and their solutions.

Continuous Function

Definition

y = f(x)

limx → af(x) = f(a)
Continuous: the graph is not disconnected.
If (limit value) = (function value),
then f(x) is continuous at x = a.
Limit (Math)
y = f(x)

limx → a-f(x) = limx → a+f(x) = f(a)
So if (left-hand limit) = (right-hand limit) = (function value),
then f(x) is continuous at x = a.
One-Sided Limits
If f(x) is continuous at every point,
then f(x) is a continuous function.
Continuous Function:
Polynomial
Exponential
Sine/Cosine
...

Example

f(x) = {x2 + 1 (x < 1)
-x + 3 (x ≥ 1)


Determine whether f(x) is continuous at x = 1.
Solution

Example

f(x) = {(x + 1)(x - 3)(x - 3) (x ≠ 3)
2 (x = 3)


Determine whether f(x) is continuous at x = 3.
Solution