Question

Let f be a function from the set of real numbers to itself. Prove that if f is differentiable at a real number c, then f is continuous at c.

Answer 1

By definition, it is a necessary condition for a function to be differentiable that it is continuous in that point

Being continuous means the function exists at that point and has limit equal to the value of the function at that point. The definition of derivative is:\[\lim_{h \rightarrow 0}\frac{ f(c+h)-f(c) }{h }\]shouldn't be continuous in "c", then the limit could not be calculated

Answer 2

So how should i format my proof though? I know that i should start with the definition of derivative but then what?

Answer 3

That is the proof, the definition of differentiable function
http://en.wikipedia.org/wiki/Differentiable_function#Differentiability_and_continuity

Answer 4

how is that a proof? I have do write the whole proof how it ends up to be continuous meaning f(x)=f(c)

Answer 5

First, dont get angry.
Second: f(x)=f(c) is not the definition of continuity
Third, here goes a rationale:

From the definition of derivative at "c" follows that when h-->0:
f(c+h)=f(c)+hf'(c) and f(c-h)=f(c)-h·f'(c) because the derivative must be the same to the left and to the right. From these expressions you can see that:
1)f(c) exists
2)f(c+h) exists and f(c-h) exists
3)h-->0 means f(c+h)-->f(c) [because f'(c) is a finite number and h-->0 we have h·f'(c)-->0], that is the limit to the right is f(c)
4)h-->0 means f(c-h)-->f(c)[for the same reason stated above], that is the limit to the left is also f(c)

So we have a function that exists in "c", has limit in "c" and the value of the limit is f(c)
That is the definition of continuity, my friend, and not f(x)=f(c)