I need a function which satisfies these conditions:
- $f(0) = 0,$
- $f(x)$ is monotonically increasing for all $x > 0,$
- for some specified $x_2 > x_1 > 0$ and $y_2 > y_1 > 0,$ $f(x_1) = y_1,$ and $f(x_2) = y_2.$
I can construct $f(x)$ easily using a piecewise linear function that passes from the origin to the points $(x_1,y_1)$ and $(x_2,y_2)$ in turn, then continues to monotonically increase at some arbitrary rate. However, I am concerned at possible effects of the non-continuous curvature. Is there a better alternative to using spline curves that might give both a smooth curve and a non-piecewise function?