Knowing if the real derivative exists The numerical derivative is valid only if the real derivative exists. Is it possible to know if the real derivative exists without using symbolic derivative, and using computer operations?
 A: From reading the comments above, I'm going to reformulate your question: given the ability to call a procedure 
float y = f(float x)
(but not to look at the code for it!), you can estimate the derivative at a point $c$ by computing
h = .00001
d = (f(c + h) - f(c) ) / h

which will give you a numerical estimate. If you make $h$ smaller, the estimate will perhaps get better, or seem to converge on some value. You'd like to know whether the (mathematical) function implemented by the code in your procedure is in fact differentiable at $c$. 
The answer is that you cannot know. Suppose that EPS is the smallest floating point number in your computer.  Consider these two functions
float y = f1(float x) :
if (abs(x) < EPS ) and (x is not 0) :
    y = 1
else
    y = 0


float y = f2(float x):
    y = 0

These return identical results for all values of $x$. But the first is an implementation of a function that's discontinuous, hence not differentiable, at 0, while the second implements a function that's certainly differentiable. But you cannot, using your computer, tell the difference. 
I hope that this is of some help to you. The short form is "nothing you can do (easily) computationally is of much help is dealing with the intricacies of real analysis."
