Animated Solution for Mathematics - Limits, Continuity and Differentiability: Contrapositive of the statement: 'If a function $f$ is differentiable at a, then it is also continuous at a', is :

Understanding the Contrapositive

Mathematics is not just about numbers and equations; it is about the elegant, unbreakable structure of logic that holds the universe of calculus together. Today, we are going to master the concept of the contrapositive, a tool that is as powerful as it is simple.

We start with a foundational truth in calculus: "If a function $f$ is differentiable at $a$, then it is also continuous at $a$." This is a conditional statement, which we can represent symbolically as $P\to Q$.

Here, $P$ is our hypothesis—the condition that the function is differentiable at $a$. $Q$ is our conclusion—the result that the function must be continuous at $a$.

Imagine the universe of all functions. Within this universe, there is a large set of functions that are continuous at $a$. Inside that set, there is a smaller, more exclusive circle of functions that are differentiable at $a$.

This is the geometric reality: every function that is differentiable is, by necessity, also continuous. If you are inside the red circle of differentiability, you are automatically inside the blue circle of continuity. There is no escape!

Now, we want to find the contrapositive. The rule is simple yet profound: the contrapositive of $P\to Q$ is $\sim Q\to \sim P$. We are essentially looking at the same truth from the opposite side.

We negate the conclusion and the hypothesis, and we swap them. Let us apply this to our calculus statement. The negation of $Q$ ($\sim Q$) is "The function $f$ is not continuous at $a$."

The negation of $P$ ($\sim P$) is "The function $f$ is not differentiable at $a$." When we put them together in the form $\sim Q\to \sim P$, we get: "If a function $f$ is not continuous at $a$, then it is not differentiable at $a$."

Why do we care about this? Because in the heat of a JEE Advanced exam, you might be asked to prove that a function is not differentiable. Instead of struggling with limits and derivatives, you can simply check for continuity.

If you find that the function is discontinuous at $a$, the contrapositive tells you immediately—without any further calculation—that it cannot be differentiable at $a$. It is a shortcut born from pure logic.

Remember, a statement and its contrapositive are like twins; they always share the same truth value. By mastering this, you are not just solving a logic problem; you are sharpening your intuition for the entire field of analysis.

Keep this logical framework in your toolkit, and you will find that even the most complex problems begin to yield to your understanding.