if a function is differentiable for all real values of x, can i assume the derivative defined on all real values of x? I think the answer is yes but i couldn't think of based explanation. if it is so, why? thank you for your help.
You are right. The definition of an everywhere differentiable function is a function which has derivative at all points.
If $x_0$ is a point in the domain of a function f, then f is said to be differentiable at $x_0$ if the derivative f ′($x_0$) exists.