Question

Assertion (A): \[ f(x)=|x| \] is differentiable at \(x=a\neq 0\) and continuous but not differentiable at \(x=0\). Reason (R): If a function is differentiable at a point, then it is continuous at that point. But the converse is not true.

• A. A is correct, R is correct, R is correct explanation of A
• B. A is correct, R is correct, but R is not correct explanation of A
• C. A is correct, R is false
• D. A is false, R is correct

Hint:

Always remember: \[ \text{Differentiable} \Rightarrow \text{Continuous} \] but \[ \text{Continuous} \nRightarrow \text{Differentiable}. \] The function \[ f(x)=|x| \] is the most common example of a function that is continuous but not differentiable at a point.

Answer 1

The Correct Option is A

Step 1: Verify Assertion A.
\(f(x)=|x|\): for \(a \neq 0\), \(f'(a)=\pm 1\) (exists), so differentiable. At \(x=0\): left derivative \(-1 \neq\) right derivative \(+1\), so continuous but not differentiable. A is correct.

Step 2: Verify Reason R.
Differentiability implies continuity (standard theorem). The converse is false (e.g., \(|x|\) at 0). R is correct.

Step 3: Assess if R explains A.
R directly justifies why continuity at 0 does not imply differentiability there, which is precisely what A illustrates. R is the correct explanation of A.
\[ \boxed{\text{Both A and R are true; R is the correct explanation of A}} \]