# Differentiability implies Continuity but Converse not True

## Derivative implies Continuity but Converse NOT True:

In this section, we will prove that if a function is differentiable at a point, then the function is continuous at that point. Its converse statement is the following: if a function is continuous, then it is not necessarily differentiable. We prove the converse statement by providing examples.

Theorem 1: If f(x) is differentiable at x=a, then it is continuous at x=a.

Proof:  Given that f(x) is differentiable at x=a.

So the derivative of f(x) at x=a is finite. In other words,

$f'(a)$ = limh→0 $\dfrac{f(a+h)-f(a)}{h}$ exists.

Now, we have

limh→0 [f(a+h)-f(a)]

= limh→0 $[\dfrac{f(a+h)-f(a)}{h} \times h]$

= limh→0 $\dfrac{f(a+h)-f(a)}{h}$ × limh→0 h

= $f'(a)$ × 0

= 0.

Thus, we obtain that

limh→0 [f(a+h)-f(a)] = 0

⇒ limh→0 f(a+h) – limh→0 f(a) = 0

⇒ limh→0 f(a+h) – f(a)=0

⇒ limh→0 f(a+h) f(a+h)=f(a)

This makes f(x) is continuous at x=a.

Thus, we have shown that if a function is differentiable at x=a then it must be continuous at x=a.

## Supporting Example

Let f(x)=x.

We will show that f(x)$i continuous and differentiable at x=2. Continuity Checking: Note that f(2)=2. We have limx→2- f(x) = limx→2- x = 2. Again, limx→2+ f(x) = limx→2+ x = 2. Thus limx→2- f(x) = limx→2+ f(x) = f(2). This shows that f(x) is continuous at x=2. Differentiability Checking: By the definition of derivative, we have$f'(2)$= limh→0$\dfrac{f(2+h)-f(2)}{h}$= limh→0$\dfrac{2+h-2}{h}$= limh→0$\dfrac{h}{h}\$

= limh→0 1

= 1.

Thus, f(x)=x is differentiable at x=2.

As f(x) is continuous and differentiable at x=2, this example supports the theorem saying if f(x) is differentiable at a point, then it is continuous at that point.

## Converse of Differentiability implies Continuity is NOT True

To show the converse of “differentiability implies continuity” is not true, we will consider the following example:

Let f(x)=|x| (the absolute value of x).

The function f(x)=|x| is continuous at x=0 but Not differentiable at x=0.