Assertion (A) Evaluation: $f(x) = x|x|$ at $x=0$

We need to check if the function $f(x) = x|x|$ is continuous and differentiable at $x=0$.

• Continuity at $x=0$:
  • The function value at $x=0$ is $f(0) = 0 \times |0| = 0$.
  • The limit from the right: $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x(x) = \lim_{x \to 0^+} x^2 = 0$.
  • The limit from the left: $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x(-x) = \lim_{x \to 0^-} -x^2 = 0$.
  • Since $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$, the function is continuous at $x=0$.
• Differentiability at $x=0$:
  • First, let's write $f(x)$ piecewise: $f(x) = \begin{cases} x^2 & \text{if } x \ge 0 \\ -x^2 & \text{if } x < 0 \end{cases}$.
  • The derivative from the right ($f'_+(0)$) is: $ f'_+(0) = \lim_{h \to 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h|h|}{h} = \lim_{h \to 0^+} \frac{h(h)}{h} = \lim_{h \to 0^+} h = 0 $
  • The derivative from the left ($f'_-(0)$) is: $ f'_-(0) = \lim_{h \to 0^-} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0^-} \frac{h|h|}{h} = \lim_{h \to 0^-} \frac{h(-h)}{h} = \lim_{h \to 0^-} -h = 0 $
  • Since $f'_+(0) = f'_-(0) = 0$, the derivative exists at $x=0$, and $f'(0) = 0$.

Therefore, Assertion (A) is correct.

Reason (R) Evaluation: Continuity and Differentiability Relationship

Reason (R) states that it is a necessary and sufficient condition that every continuous function is derivable at any point.

• This statement is incorrect.
• Continuity at a point is a necessary condition for differentiability at that point, meaning if a function is differentiable, it must be continuous.
• However, continuity is not sufficient to guarantee differentiability.
• A common counterexample is the function $g(x) = |x|$. This function is continuous at $x=0$ (since $g(0)=0$ and $\lim_{x \to 0} |x| = 0$), but it is not differentiable at $x=0$ because the left-hand derivative ($ -1 $) does not equal the right-hand derivative ($ +1 $).

Therefore, Reason (R) is incorrect.

Conclusion

Based on the analysis:

• Assertion (A) is correct.
• Reason (R) is incorrect.

This corresponds to the option stating that (A) is correct but (R) is not correct.