Continuity and Differentiability
Relationship between Continuity and Differentiability
GRB_1000_MCQ
Grade Class 12
Question:
Which of the following statements are true?
If $\lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}$ exists, then $f$ is differentiable at $a$.
If $f$ is continuous at $a$, then $f$ is differentiable at $a$.
If $\lim_{x \to a} f(x)$ exists, then $f$ is differentiable at $a$.
If $f$ is differentiable at $a$, then $\lim_{x \to a} f(x) = f(a)$.
Step-by-Step Solution
Key Concept: The key idea here is understanding the precise definitions of differentiability, continuity, and the existence of a limit at a point, along with the correct hierarchical implications between them: differentiability at a point implies continuity at that point, which further implies the existence of the limit of the function at that point.
Step 1: Analyze option (a). The definition of differentiability at $a$ is exactly that $\lim_{x \to a} \dfrac{f(x)-f(a)}{x-a}$ exists. So if this limit exists, $f$ is differentiable at $a$. Option (a) is TRUE.
Step 2: Analyze option (b). Continuity does not imply differentiability. A classic counterexample is $f(x) = |x|$ at $a = 0$: it is continuous but not differentiable. Option (b) is FALSE.
Step 3: Analyze option (c). The existence of $\lim_{x \to a} f(x)$ does not even imply continuity (the limit may not equal $f(a)$), let alone differentiability. Option (c) is FALSE.
Step 4: Analyze option (d). Differentiability at $a$ implies continuity at $a$, which means $\lim_{x \to a} f(x) = f(a)$. Option (d) is TRUE.
Correct Answer: 1, 4