Relationship: Differentiability and Continuity

Hello JEE aspirants! In this lesson, we'll explore the fascinating connection between differentiability and continuity. These two concepts are fundamental to calculus and play a crucial role in solving many JEE problems. Understanding their relationship will give you a powerful edge in tackling complex questions.

1. Theorem: Differentiable ⟹ Continuous

This theorem states that if a function $f(x)$ is differentiable at a point $x = a$ , then it must be continuous at that point. In simpler terms, if you can find the derivative of a function at a specific point, then the function is guaranteed to be continuous there.

Intuition: Think of differentiability as requiring a "smooth" curve. To have a derivative, you need to be able to draw a tangent line at a point. This is only possible if there are no breaks or jumps in the graph, which is what continuity ensures. Imagine trying to draw a tangent line on a broken curve; it's impossible!

\text{If } f \text{ is differentiable at } x = a, \text{ then } f \text{ is continuous at } x = a

Explanation: Let's understand why this is true. If $f(x)$ is differentiable at $x = a$ , then the limit $f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$ exists. Now, let's rewrite $f(x) - f(a)$ as: $f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)$ Taking the limit as $x \rightarrow a$ : $\lim_{x \rightarrow a} [f(x) - f(a)] = \lim_{x \rightarrow a} \left[ \frac{f(x) - f(a)}{x - a} \cdot (x - a) \right]$ $\lim_{x \rightarrow a} [f(x) - f(a)] = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a} \cdot \lim_{x \rightarrow a} (x - a)$ Since $f$ is differentiable at $x = a$ , the first limit on the right-hand side is $f'(a)$ , and the second limit is 0: $\lim_{x \rightarrow a} [f(x) - f(a)] = f'(a) \cdot 0 = 0$ This implies that: $\lim_{x \rightarrow a} f(x) = f(a)$ which is the definition of continuity at $x = a$ .

2. Converse is false: Continuous ⇏ Differentiable

The converse of the above theorem is not true. Just because a function is continuous at a point, it doesn't necessarily mean it's differentiable at that point. This is a crucial distinction!

Intuition: Continuity only requires the function to be "connected" at a point. Differentiability requires the function to be smooth. You can have a connected curve that has sharp corners or cusps, where you can't draw a unique tangent line. Those points are continuous but not differentiable.

\text{If } f \text{ is continuous at } x = a, \text{ it does NOT imply that } f \text{ is differentiable at } x = a

3. Examples of continuous but non-differentiable functions

Let's look at some classic examples:

$f(x) = |x|$ at $x = 0$ : The absolute value function is continuous everywhere, including at $x = 0$ . However, it has a sharp corner at $x = 0$ . The left-hand derivative is -1, and the right-hand derivative is +1. Since the left and right-hand derivatives are not equal, the function is not differentiable at $x = 0$ .
$f(x) = \sqrt[3]{x}$ at $x = 0$ : This function is continuous everywhere. However, it has a vertical tangent at $x = 0$ . The derivative approaches infinity as $x$ approaches 0, so the function is not differentiable at $x = 0$ .

4. Sharp corners, cusps, vertical tangents

These are the usual suspects when looking for points where a function is continuous but not differentiable.

Sharp corners: As seen with $f(x) = |x|$ . The function changes direction abruptly.
Cusps: A cusp is similar to a sharp corner but more extreme. An example is $f(x) = x^{2/3}$ .
Vertical tangents: The tangent line becomes vertical, meaning the derivative approaches infinity, as seen with $f(x) = \sqrt[3]{x}$ .

\text{Contrapositive: If } f \text{ is not continuous at } x = a, \text{ then } f \text{ is not differentiable at } x = a

Explanation: The contrapositive is logically equivalent to the original statement. If a function has a discontinuity (jump, hole, vertical asymptote) at $x = a$ , it cannot have a derivative at that point. This is because the limit defining the derivative won't exist.

Tip: Always check for continuity before attempting to find the derivative. If a function is discontinuous, you know immediately that it's not differentiable.

Common Mistake: Assuming that continuity implies differentiability. Remember the converse is NOT true!

JEE Specific Tricks: When solving JEE problems, look for piecewise functions. These are often designed to be continuous but not differentiable at the points where the pieces join. Pay close attention to the left-hand and right-hand derivatives at these points.

Understanding the relationship between differentiability and continuity is not just about memorizing theorems; it's about developing a deeper intuition for how functions behave. Keep practicing with examples, and you'll master this concept in no time!