If we can plot the graph of a function without lifting our pen, we can say that it is a continuous function. If we lift our pen to plot a certain part of a graph, the function is discontinuous. In this article, we discuss the Theorems of Continuity. We explain the continuity of f(x)±g(x), f(x).g(x), depending upon whether f(x) and g(x) are continuous or discontinuous functions.
Continuous Function
A function, f(x), is said to be continuous at a point x = a, if and only if
- f(a) exists
- \(\begin{array}{l}\lim_{x\to a}f(x)=f(a)\end{array} \)
- \(\begin{array}{l}\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=f(a)\end{array} \). i.e LHL = RHL = f(a)
Theorems of Continuity are as follows.
Theorem 1
Let f(x) and g(x) be continuous functions at x = a, then
a. (f(x)+ g(x)) is continuous at x = a
b. (f(x)- g(x)) is continuous at x = a
c. (f(x). g(x)) is continuous at x = a
d. (f(x)/ g(x)) is continuous at x = a, if g(a) is not equal to zero.
If g(a) = 0, then (f(x) / g(x)) is discontinuous at x = a.
Theorem 2
If f(x) is continuous and g(x) is discontinuous at x = a, then the functions defined by
a) f(x) + g(x) will be discontinuous at x = a.
b) f(x) – g(x) will be discontinuous at x = a.
c) f(x).g(x) and f(x)/g(x), (g(a) ≠ 0) may be continuous at x = a.
Theorem 3
If g(x) is a continuous function at x = a and function f(x) is continuous at g(a), then the composition fog is continuous at x = a.
Theorem 4
If limx→ a g(x) = L and if f(x) is continuous at x = L, then limx→ a f(g(x)) = f(limx→ a g(x)) = f(L).
Also Read
Limits Continuity and Differentiability
Solved Examples of Theorems of Continuity
Example 1:
The values of x at which function f(x) = (x-2)/(2x2+2x-4)(x4+5) is discontinuous is
a) x = 1, x = -2
b) x = -1, x = 2
c) x = -1, x = -2
d) None of the above
Solution:
Given f(x) = (x-2)/(2x2+2x-4)(x4+5)
f(x) is discontinuous => 2x2+2x-4 = 0 (Here, x4+5 will be always positive, so it cannot be zero.)
=> x2+x-2 = 0
=> (x-1)(x+2) = 0
=> x = 1, x = -2
Hence, option a is the answer.
Example 2:
Find the value of k, so that the function is continuous at x = 0.
a) 1
b) -1
c) 0
d) None of the above
Solution:
Given
limx→ 0f(x) = f(0) (since f(x) is continuous at x = 0)
limx→ 0f(x) = limx→ 0 1-cos 4x
= 1-1
= 0
So k = 0
Hence, option c is the answer.
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Frequently Asked Questions
Give the conditions for a function to be continuous at x = a.
A function f(x) is said to be continuous at a point x = a, if f(a) exists, lim x→a f(x) = f(a) and lim x→ a- f(x) = lim x→ a+ f(x) = f(a).
Give two examples of a continuous function.
The trigonometric functions sin x and cos x are continuous.
List two properties of continuous function.
Let f(x) and g(x) be two functions, continuous at x = a. Then, f(x) + g(x) will be continuous at x = a.
f(x) – g(x) will be continuous at x = a.
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