Find function satisfying specific conditions

Question

All:

I am wondering if there is function can emulate a curve that satisfies:

[1] Passing (0,0) (90,50) (100,100)
[2] Continuous and Smooth (left-lim == right lim)
[3] Keep rising(100>=x1>x2>=0 then y1>=y2)

I feel it does not exist( or if you can prove it exist and solution is more than one, just give me any one of them), but I do not know how to prove it

Thanks

Answer 1

We can fit it to a power function of the form $y = ax^k$ for some $a$ and $k$. Solving the system yields:

$$ y = \frac{50}{90^{\log_{10/9}2}}x^{\log_{10/9}2} $$

Here's a plot from Wolfram|Alpha.

Answer 2

Use line segments. Take: $$y(x) = \begin{cases} x, \text{ if }x \leq 0 \\ 5x/9, \text{ if } 0 < x \leq 90 \\ 50 + 5(x-90), \text{ if } x > 90\end{cases}$$ The function is continuous, increasing, and passes through all points. Do notice, though, that "smooth" is often used for referring to differentiability.

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