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.
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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 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.