Antiderivatives and Riemann integrals: some observations
The fundamental theorem of calculus sets a link between the
concept of antiderivative and of Riemann integral:

the Riemann integral is an
antiderivative of the continuous function f;

given an antiderivative G(x) of a continuous
function f we have: .
Despite this important link, the two concepts remain quite
different: the algorithm used to calculate antiderivatives
produces a class of functions; the one used to calculate Riemann
integrals produces a real number.
Besides this we want to point out the following facts.

The set of all integral functions of a given continuous
function f is, in general, a subset of the set of
all antiderivatives. An integral function is always zero at
the starting point: , while an
antiderivative may be always strictly positive. For example
the function F(x)=x^{2}+1 is an
antiderivative of f(x)=2x, but it is
nowhere zero: this means that it can't be an integral
function.

A function that has a jump discontinuity can't have
antiderivatives, while it may be integrable. The simplest
examples are step functions.

The integral functions of a non continuous function usually
are not antiderivatives. For example , but the function x is not an
antiderivative, as it is not differentiable in zero. Observe
that this equality is an alternative definition of the
absolute value function.

An unbounded function can't be Riemann integrable, while
it may have antiderivatives. For an example try to find the
derivative, f(x), of . As f is unbounded it is certainly not
Riemann integrable, but obviously it has an antiderivative,
the function F itself!
copyright 2000 et seq. maddalena falanga & luciano battaia
first published on january 07 2003  last updated on september 01
2003