Sets, Relations & Functions
Logic and Statements
Grade 11
Question:
<p>Contrapositive of the statement 'If a function \(f\) is differentiable at \(a\), then it is also continuous at \(a\)', is</p>
<p>(a) If a function \(f\) is not continuous at \(a\), then it is differentiable at \(a\)</p>
<p>(b) If a function \(f\) is continuous at \(a\), then it is differentiable at \(a\)</p>
<p>(c) If a function \(f\) is continuous at \(a\), then it is not differentiable at \(a\)</p>
<p>(d) If a function \(f\) is not continuous at \(a\), then it is not differentiable at \(a\)</p>
Step-by-Step Solution
Key Concept: The contrapositive of 'If P then Q' is 'If not Q then not P', and it is logically equivalent to the original statement.
<p>The contrapositive of 'If P, then Q' is 'If not Q, then not P'. For the statement 'If a function \(f\) is differentiable at \(a\), then it is continuous at \(a\)', the contrapositive is 'If a function \(f\) is not continuous at \(a\), then it is not differentiable at \(a\)'.</p><p>∴ Answer is (d).</p>
Correct Answer: D