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**algebraic topology** Well, here’s a modification of CaptainBlack’s suggestion.

Join the first two points ($\displaystyle (a,A)$ and $\displaystyle (b,B)$) by a straight line. This will have gradient $\displaystyle \frac{B-A}{b-a}$. Then join $\displaystyle (b,B)$ to $\displaystyle (c,C)$ by the function $\displaystyle y=f(x)=C_1\mathrm{e}^{C_2x}+C_3$, where $\displaystyle f(b)=B$, $\displaystyle f(c)=C$, and $\displaystyle f'(b)=\frac{B-A}{b-a}$.

Hopefully that might makes things a bit simpler.

By the way, CaptainBlack, the condition $\displaystyle C_1,C_2>0$ will make $\displaystyle f(x)$ convex, and this will only work if $\displaystyle (b,B)$ lies below the straight line joining $\displaystyle (a,A)$ and $\displaystyle (c,C)$. If $\displaystyle (b,B)$ lies above that line, then we need a concave function, which means we need both $\displaystyle C_1$ and $\displaystyle C_2$ to be negative ($\displaystyle C_2<0$ to make the function concave and $\displaystyle C_1<0$ to make it increasing).