# [Maxima] unit_step() and integrate()

Rupert Swarbrick rswarbrick at gmail.com
Mon Jan 16 04:10:11 CST 2012

```"Richard Hennessy" <rich.hennessy at verizon.net> writes:
> I meant the indefinite integral of the square wave function is
> continuous everywhere.  Sorry.  In general if a function involves step
> functions the integral in usually continuous, even though the original
> function is not.
>
> Rich

*sigh*

If f is a function from the real line to itself and you define a new
function F by

F(x) = integrate (f(y), y, a, x)

for any choice of a, then F is called the "indefinite integral" of f. A
theorem, which is sort of trivial once you've set up all the machinery
for Riemann integration, says that F is always continuous (unit steps or
no!)

Occasionally, people talk of "functions" like the Dirac delta function
δ(x). This isn't, strictly speaking, a function R→R because it doesn't
evaluate to a number at x=0. Rather it is defined by

integrate (δ(x), x, a, b) = 1 if a<0<b and 0 otherwise

An indefinite integral for δ looks like a unit step function (and is a
proper honest-to-god function). This isn't continuous.

If you're interested to know more about things like δ and more general
such objects, look up the theory of "distributions" (which is the more
general name for such objects) or "weak derivatives".

I hope this makes things a bit clearer,

Rupert
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