Key Concepts and Formulas

• Conditional Statement: A conditional statement is a statement of the form "If p, then q," denoted as $p \rightarrow q$, where p is the hypothesis and q is the conclusion.
• Contrapositive: The contrapositive of a conditional statement $p \rightarrow q$ is $\sim q \rightarrow \sim p$, where $\sim$ denotes negation. In words, the contrapositive is "If not q, then not p."
• Negation: The negation of a statement reverses its truth value. If p is a statement, then $\sim p$ is "not p."

Step-by-Step Solution

Step 1: Identify the hypothesis and conclusion.

The given statement is "If a function f is differentiable at a, then it is also continuous at a." Let p be the statement "a function f is differentiable at a." Let q be the statement "a function f is continuous at a." Therefore, the given statement can be represented as $p \rightarrow q$.

Step 2: Determine the negation of the hypothesis and conclusion.

We need to find $\sim p$ and $\sim q$. $\sim p$: "a function f is not differentiable at a." $\sim q$: "a function f is not continuous at a."

Step 3: Form the contrapositive statement.

The contrapositive of $p \rightarrow q$ is $\sim q \rightarrow \sim p$. Substituting the negations we found in Step 2, we get: "If a function f is not continuous at a, then it is not differentiable at a."

Step 4: Match the contrapositive with the given options. The contrapositive statement "If a function f is not continuous at a, then it is not differentiable at a" matches with option (C).

Common Mistakes & Tips

• Remember the correct order for the contrapositive: it is $\sim q \rightarrow \sim p$, not $\sim p \rightarrow \sim q$. Reversing the order is a common mistake.
• Be careful when negating statements, especially those involving quantifiers (e.g., "all," "some"). In this case, the negations are straightforward, but in other problems, they can be more complex.
• Understanding the relationship between a statement and its contrapositive is crucial. A statement and its contrapositive are logically equivalent, meaning they have the same truth value.

Summary

We identified the hypothesis and conclusion of the given statement, negated both, and then formed the contrapositive statement. This process led us to the contrapositive: "If a function f is not continuous at a, then it is not differentiable at a," which corresponds to option (C). However, the stated "Correct Answer" is (A), indicating an error in the original answer key. The correct contrapositive is derived by negating the "continuous" part to get "not continuous" and negating the "differentiable" part to get "not differentiable" then reversing the order. This gives "If not continuous, then not differentiable."

The final answer is \boxed{C}.