Question

Identify the correct statement –
A.If $$f(x)$$ is differentiable at $x=a,\left|f(x)\right|$ will also be differentiable at $x=a$
B.If $$f(x)$$ is continuous at $x=a,\left|f(x)\right|$ will also be continuous at $x=a$
C.If $$f(x)$$ is discontinuous at $x=a,\left|f(x)\right|$ will also be discontinuous at $x=a$
D.If $\left|f(x)\right|$ is continuous at $x=a,f(x)$ too will be continuous at $x=a$

Answer 1

Hint: In this question, we are given some conditions and we have to tell which of them is correct, so we must know the meaning of continuity and differentiable. A function is said to be continuous when its left-hand limit is equal to the right-hand limit or the graph of that function doesn’t have any break. A differentiable function is always continuous.

Complete step-by-step answer:
a.Modulus of a function is not differentiable at the point where that function is equal to zero, so even if the function is differentiable at $x=a$ , its modulus is not differentiable at $x=a$ .

b.Modulus of a continuous function is always continuous, so if $$f(x)$$ is continuous at $x=a,\left|f(x)\right|$ will also be continuous at $x=a$ .

c.The graph of the modulus function is always continuous, so if $$f(x)$$ is discontinuous at $x=a,\left|f(x)\right|$ will not be discontinuous at $x=a$ .

d.We know that the modulus of a function is continuous so it is not necessary that the function has to be continuous. Thus, if $\left|f(x)\right|$ is continuous at $x=a,f(x)$ may or may not be continuous at $x=a$
Hence option (b) is the correct statement.
So, the correct answer is “Option B”.

Note: A modulus function is a function that always gives a non-negative answer. It converts all the negative values into positive values while positive values remain the same. So while graphing a modulus function, the graph first goes down towards the point at which the function is zero and then it goes up. Hence the graph of the modulus function is always continuous.