Man, that is riddled with errors.

There is
She oscillated violently, lost all sense of directrix, and went completely divergent.
followed by
"Of course not," Polly replied quite properly, "I'm absolutely convergent."
which is inconsistent. And besides, polynomial functions always have a finite number of power series terms, so they cannot diverge.

And here is another problem:
Curly removed her discontinuities.
her mother noticed that she was no longer piecewise continuous
This would imply that it used to be piecewise continuous, but then there are no discontinues to remove. And a polynomial never has discontinuities or jumps anyway.
Quite suddenly two branches of a hyperbola touched her at a single point.
I will assume from this that curlypi(x) is some kind of hyperbolic function.
That smooth operator, Curly Pi
A degenerate hyperbola is not exactly smooth.

If they mean the symbol itself, this is what a curly pi looks like:
v. how is that smooth?
erf
The only special function they had in there was something dinky like erf?!

he performed Runge-Kutta on her.
Why the heck would you do that with a polynomial?

He integrated by parts. He integrated by partial fractions
not much point in doing all that for a stupid polynomial...

Curly went on operating until he completely satisfied her hypothesis, then he exponentiated and became completely orthogonal.
exp(curlypi(x)) certainly does not satisfy orthogonality relations...
When Polly got home that night, her mother noticed that she was no longer piecewise continuous, but had been truncated in several places.
Then that's not a polynomial anymore; those must be fully continuous and analytic everywhere.
Finally she went to l'Hôpital
l'hopital was a mathematician, but he did not have anything to do with surds or CEDN functions...
There are other problems, but I don't have the time time to go through it all now.
