# 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.limx → af(x) = f(a)

If (limit value) = (function value),

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

Limit (Math)

y = f(x)

limx → a

So if (left-hand limit) = (right-hand limit) = (function value),limx → a

^{-}f(x) = limx → a^{+}f(x) = f(a)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) = {x

-x + 3 (x ≥ 1)

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

Solution ^{2}+ 1 (x < 1)-x + 3 (x ≥ 1)

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

limx → 1

= limx → 1

= (1

= 1

= 1 + 1

= 2

limx → 1

= limx → 1

= -(1

= -1 + 3

= 2

f(1) = -1 + 3

= 2

limx → 1

∴ f(x) is continuous at x = 1.

^{-}f(x)= limx → 1

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

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

^{2}+ 1= 1 + 1

= 2

limx → 1

^{+}f(x)= limx → 1

^{+}(-x + 3)= -(1

^{+}) + 3= -1 + 3

= 2

f(1) = -1 + 3

= 2

limx → 1

^{-}f(x) = limx → 1^{+}f(x) = f(1)∴ f(x) is continuous at x = 1.

Close

### Example

f(x) = {(x + 1)(x - 3)(x - 3) (x ≠ 3)

2 (x = 3)

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

Solution 2 (x = 3)

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

limx → 3f(x)

= limx → 3(x + 1)(x - 3)(x - 3)

= limx → 3(x + 1)

= 3 + 1

= 4

f(3) = 2

limx → 3f(x) ≠ f(3)

∴ f(x) is not continuous at x = 3.

= limx → 3(x + 1)(x - 3)(x - 3)

= limx → 3(x + 1)

= 3 + 1

= 4

f(3) = 2

limx → 3f(x) ≠ f(3)

∴ f(x) is not continuous at x = 3.

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