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)=x2+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