Question

Given below are two statements:
Statement I: A function which is differentiable at a point must also be continuous at that point
Statement II: A function which is continuous at a point is not necessarily differentiable at that point
In light of the above statements, choose the most appropriate answer from the options given below

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Hint:

Think of differentiability as a stronger property than continuity.
Differentiability implies continuity, but continuity does not imply differentiability.
Visual metaphor: A continuous path can have sharp corners (like \(|x|\)), but a differentiable path must be completely smooth.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Continuity and differentiability are fundamental properties of real-valued functions in calculus.
- A function is continuous at a point if it has no breaks, jumps, or holes there.
- A function is differentiable at a point if it has a well-defined, unique tangent line at that point (i.e., the limit of the difference quotient exists).

Step 2: Detailed Explanation:
Let us analyze both statements:
- Statement I: "A function which is differentiable at a point must also be continuous at that point."
This is a standard theorem in calculus.
If a function \(f(x)\) is differentiable at \(x = c\), then by definition:
\[ \lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c) \] We can write:
\[ f(x) - f(c) = \frac{f(x) - f(c)}{x - c} \cdot (x - c) \] Taking the limit as \(x \to c\):
\[ \lim_{x \to c} [f(x) - f(c)] = \lim_{x \to c} \left[ \frac{f(x) - f(c)}{x - c} \right] \cdot \lim_{x \to c} (x - c) = f'(c) \cdot 0 = 0 \] This implies \(\lim_{x \to c} f(x) = f(c)\), which is the definition of continuity. Thus, Statement I is correct.
- Statement II: "A function which is continuous at a point is not necessarily differentiable at that point."
This is also correct.
The classic counterexample is the absolute value function:
\[ f(x) = |x| \] At \(x = 0\), the function is continuous because \(\lim_{x \to 0} |x| = 0 = f(0)\).
However, the left-hand derivative is \(-1\) and the right-hand derivative is \(+1\). Since these limits do not match, the derivative does not exist at \(x = 0\). Thus, Statement II is correct.

Step 3: Final Answer:
The correct option is (A).