Theorem 0.1. If a function f(x) is differentiable at a point x = c in its domain, then f(c) is continuous at x = c. 

Note that the converse is definitely not true, as, for example, f(x) = |x| is continuous at x = 0, but not differentiable there. Note also that, for a function f(x) to be continuous at x = c,we must have

 lim x→c f(x) = f(c)

But we can also write this as limx→c f(x) − f(c) = 0. 

This will be useful.

Proof. Assume we have a function f(x) that is differentiable at a point x = c in its domain. Then the limit

                                      

                                           

                                                                        f(x) − f(c)

                                         f' (c) = lim x→c     -------------

                                                                            x - c                          

exists. Knowing this, we calculate lim x→c f(x) − f(c) as follows (we would like it to be 0):

  

Let 

f$$�$$

 be the differentiable function at 

x=a$$�=�$$

.
Then according to the basic definition of the differentiation, Differentiation of a function is equals to

f′(c)=limx→af(x)−f(a)x−a$${ }^{}{\displaystyle \frac{\mathrm{<span\; style="font-family:\; "Open\; Sans",\; sans-serif;\; word-spacing:\; normal;"><br> </span>}}{}}$$

We know that the for a function to be continuous at a point it must satisfy the equation

 lim  x→a f(x)=f(a)$$\underset{ }{<br>}$$

We can write the above equation as

                       

⇒limx→a(f(x)−f(a))=0     ..........  (2)

     $$$$