Well, it's my cat which inspired me today . The goal was :find a curve using elementary function which looks like my cat and I have found this :
let $0<x<1$ the cat's curve is defined by the following function : $$f(x)=(1-x)^{\cos^2\Big(\frac{1}{x}\Big)}+x^{\cos^2\Big(\frac{1}{1-x}\Big)}-1$$
The graph looks like so:
As far as I remenber it recall me a little bit the Cantor function in the neightborhood of zero or one . Furthermore there is a big problem with the derivatives but I don't go further than the first .Maybe it's a little bit fractal .
Do you know some others interesting properties of this curve ? Exists there others curve of this kind ?
Thanks in advance cheers .:-)
Update:
As heropup make a good remark I propose to prove that :
$$\int_{0}^{1}f(x)dx<\frac{2}{3}$$
Moreover if we look at the graph of one summand I think it's hard to use taylor series (even unsuable maybe).So I don't know what tools use for.
so that the areas above and below the parabola may cancel.