JEE Main · MathematicsHard

$Identify the contrapositive of the statement: If a function f is differentiable at x = a, then f is continuous at x = a .$

A. $If f is continuous at x = a, then f is differentiable at x = a$
B. $If f is not continuous at x = a, then f is not differentiable at x = a$
C. $If f is not differentiable at x = a, then f is not continuous at x = a$
D. $If f is continuous at x = a, then f is not differentiable at x = a$

Show correct answer & step-by-step solution

Correct answer: B — $If f is not continuous at x = a, then f is not differentiable at x = a$

Solution

$Let p be 'f is differentiable at x = a' and q be 'f is continuous at x = a'.$
$The statement is p ⟹ q .$
$The contrapositive is defined as \neg q ⟹ \neg p .$
$Substituting the propositions, we get 'If f is not continuous at x = a, then f is not differentiable at x = a'.$
$Hence the answer is (B).$

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