Question

Given below are two statements:

Assertion (A):
The function \(f(x)=x|x|\) is continuous at \(x=0\) and derivative of \(f\) also exists at \(x=0\).
Reason (R):
It is necessary and sufficient that every continuous function is derivable at any point.

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Differentiability always implies continuity, but continuity does not always imply differentiability.

Answer 1

The Correct Option is C

Concept:
A function may be continuous at a point but it is not always necessary that it must be differentiable at that point. Differentiability is a stronger condition than continuity.

Step 1: Write the given function. \[ f(x)=x|x| \] Now, we remove the modulus sign by considering two cases. For \(x\geq 0\), \[ |x|=x \] so, \[ f(x)=x\cdot x=x^2 \] For \(x<0\), \[ |x|=-x \] so, \[ f(x)=x(-x)=-x^2 \] Therefore, \[ f(x)= \begin{cases} x^2, & x\geq 0\\ -x^2, & x<0 \end{cases} \]

Step 2: Check continuity at \(x=0\). \[ \lim_{x\to 0^-}f(x)=\lim_{x\to 0^-}(-x^2)=0 \] \[ \lim_{x\to 0^+}f(x)=\lim_{x\to 0^+}x^2=0 \] Also, \[ f(0)=0 \] Thus, \[ \lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)=f(0) \] So, \(f(x)\) is continuous at \(x=0\).

Step 3: Check differentiability at \(x=0\). For \(x<0\), \[ f'(x)=\frac{d}{dx}(-x^2)=-2x \] For \(x>0\), \[ f'(x)=\frac{d}{dx}(x^2)=2x \] At \(x=0\), \[ \lim_{x\to 0^-}f'(x)=0 \] and \[ \lim_{x\to 0^+}f'(x)=0 \] Hence, derivative exists at \(x=0\). Therefore, Assertion (A) is correct.

Step 4: Check Reason (R).
Reason says that every continuous function is differentiable. This is false. Every differentiable function is continuous, but every continuous function need not be differentiable. For example, \[ f(x)=|x| \] is continuous at \(x=0\), but not differentiable at \(x=0\). Thus, Assertion is correct but Reason is not correct. \[ \therefore \text{Correct Answer is (C)} \]