GATE EC 2016 Set 1 — Question 13

• Analysis: This statement is false. Continuity at a point does not guarantee differentiability.
• Counterexample: Consider the absolute value function $f(x) = |x|$ at $x_0 = 0$.
• The function is continuous at $x = 0$ because:

• However, the left-hand derivative at $x = 0$ is $-1$ and the right-hand derivative is $+1$. Since the one-sided derivatives are not equal, $f(x)$ is not differentiable at $x = 0$.
• Thus, P is false.

• Analysis: This statement is true. As demonstrated by the counterexample $f(x) = |x|$ at $x_0 = 0$, a function can be continuous at a point without being differentiable there.
• Thus, Q is true.

• Analysis: This is a fundamental theorem in calculus. If a function $f(x)$ is differentiable at $x = x_0$, then the limit:

• We can write:

• Taking the limit as $x \to x_0$ on both sides:

• This satisfies the definition of continuity at $x = x_0$. Therefore, differentiability implies continuity.
• Thus, R is true.

Conclusion

• P is false.
• Q is true.
• R is true.