W. Rudin, Principles of Mathematical Analysis.

W. Fleming, Functions of Several Variables.

For the set of real numbers R, a function

f: R --> R

x in R, d in R, function f is differentiable at x with value d provided for h in R

lim_{h --> 0} (f(x + h) - f(x))/h = d

In this case we write

f'(x) = df(x)/dx = d

In that case, in particular,

lim_{h --> 0} (f(x + h) - f(x)) = 0

so that f is continuous at x.

Differentiation is important in physics, e.g., in Newton's second law.