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-f(x) = limx → a+f(x) = f(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) = {x2 + 1 (x < 1)
-x + 3 (x ≥ 1)
Determine whether f(x) is continuous at x = 1.
Solution -x + 3 (x ≥ 1)
Determine whether f(x) is continuous at x = 1.
limx → 1-f(x)
= limx → 1-(x2 + 1)
= (1-)2 + 1
= 12 + 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.
= limx → 1-(x2 + 1)
= (1-)2 + 1
= 12 + 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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