Be prepared, for you are about to see some of the scariest functions this Halloween
Now behold, a function that can be graphed as two lines, with nothing inbetween
What sorcery is this, you might ask!
Well defining such a function is a rather simple task
Set the value at the irrationals to be zero, so none
And at the rationals one
The irrationals are dense, so there will be a line at y equals zero
But so are the rationals, and they form a line above straight as an arrow
More than a scary graph, this function has
Its nowhere continuous, no where smooth, the total opposite of jazz
Indeed, for any delta you choose, no matter how small
You shall see that the definition of continuity will fall
Choose a small enough epsilon, and you shall see
Its neighborhood must hold both zero and one, so Q.E.D
But this is not the only function we have here today
Behold this one, continuous in its on unique way
On the irrationals, it’s continuous, on the rationals not
‘How is it possible?’, you may have thought
On the irrationals it is zero, the hard part comes later
As for each rationals its value is the lowest possible denominator
Say a rational can be written in the most reduced form as a/c
Then 1/c, shall the functions value at that point be
For 0.4, we get a fifth for 0.56, one over twenty five
As the number gets more complicated, the value takes a dive
To end this show, here is the engineer’s nightmare
A continuous function, that is differentiable nowhere
“All continuous functions have a derivative” some people assumed
But then came Weierstrass and made the argument doomed
Let b be a positive odd integer greater than five
For a small positive a, the function will come alive
Just assume that a times b, minus one
Is greater than three halves pi, we are almost done
Take the series of n from zero to infinity
Of a to the n, times the cosine of the following trinity:
b to the n, times x, times pi
And behold a function with the power to petrify
Another engineer’s nightmore: f(x) = e^(-1/x^2) if x!=0, and f(0) = 0. Infinitely differentiable, with the simple power series expansion 0 + 0*x + 0*x^2 + 0*x^3 ….so the power series converges everywhere, but not to f(x) at any x except 0.