by Richard Burke-Ward
Last Updated September 21, 2018 05:20 AM

Is it possible to create an algebraic function that is smooth and continuous (i.e., a function in the form $f(x)$ using algebraic functions, with no curly braces that stipulate different behaviour for different domains of $x$, and not piecewise-continuous, smoothly continuous even in the limit) that has the following properties:

- Function is odd
- $\int_{0}^a f(x) dx=0$
- $\int_{a}^b f(x) dx=1$
- $\int_{b}^\infty f(x) dx=0$
- $\frac{d}{dx}f(x)=0, x=(a,b)$
- $\frac{d}{dx}f(x)=0$ just once in the interval between (but excluding) $x=a$ and $x=b$

The result would be a curve with vaguely sinusoidal behaviour: a single (i.e., non-oscillating) negative 'blip' between $x=(-b,a)$ and a positive 'blip' between $x=(a,b)$, and for other values of $x$, the curve either oscillates or flat-lines, but integrates to $0$ across the ranges specified.

I suspect this requires some form of Fourier analysis, and I am sure that the solution will require complex analysis, which is fine.

I understand (to some degree!) complex numbers. But Fourier analysis is something I simply don't know how to do. I just want to find such a curve so I can play with it :-)

**Edits:**

If the answer is that it's impossible, that's useful info too.

Added stipulation that the function should be smoothly continuous even in the limit.

If behaviour is different in the complex plane, that's also fine. So long as it's continuous, and the real part has the behaviour above.

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