Continuity and Differentiability · Class 12 · CBSE · Science Mathematics
Prove that if a function f is differentiable at x = a, then it is continuous at x = a.
Practice question from Continuity and Differentiability (Class 12 · CBSE · Science Mathematics), with the correct answer and a step-by-step solution.
medium · Subjective · Q-74C1D5
Prove that if a function f is differentiable at x = a, then it is continuous at x = a.
Answer: Since f is differentiable at x = a, the limit lim_{x→a} [f(x) − f(a)]/(x − a) exists and is finite. Writing f(x) − f(a) = (x − a)[(f(x) − f(a))/(x − a)] and taking x → a gives lim_{x→a}[f(x) − f(a)] = 0, so lim_{x→a} f(x) = f(a). Therefore, f is continuous at x = a.
Solution
Differentiability gives a finite limit for the difference quotient. Multiplying it by , which tends to , shows that ; therefore, .
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