**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.

**If f(x) is differentiable at x=a, then it is continuous at x=a.**

__Theorem 1:____Proof:__Given that f(x) is differentiable at x=a.

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

$f'(a)$ = lim

_{h→0}$\dfrac{f(a+h)-f(a)}{h}$ exists.Now, we have

lim

_{h→0 }[f(a+h)-f(a)]= lim

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

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

= 0.

Thus, we obtain that

lim

_{h→0 }[f(a+h)-f(a)] = 0⇒ lim

_{h→0 }f(a+h) – lim_{h→0}f(a) = 0⇒ lim

_{h→0 }f(a+h) – f(a)=0⇒ lim

_{h→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.

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**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 lim

_{x}_{→2- }f(x) = lim_{x}_{→2-}x = 2.Again, lim

_{x}_{→2+}f(x) = lim_{x}_{→2+}x = 2.Thus lim

_{x}_{→2-}f(x) = lim_{x}_{→2+}f(x) = f(2).This shows that f(x) is continuous at x=2.

__Differentiability Checking:__

By the definition of derivative, we have

$f'(2)$ = lim

_{h→0}$\dfrac{f(2+h)-f(2)}{h}$= lim

_{h→0 }$\dfrac{2+h-2}{h}$= lim

_{h→0 }$\dfrac{h}{h}$= lim

_{h→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.