Intersecting points of increasing convex functions

Question

Can two increasing and differentiable convex functions agree exactly on a countable set of cardinality greater than two?

Answer 1

Let $$f_\pm(x):=e^x(1\pm c\sin x)$$ for some $c\in(0,1/2]$ and all real $x$.

Then $f'_\pm(x)=e^x(1\pm c(\cos x+\sin x))>0$ and $f''_\pm(x)=e^x(1\pm 2c\cos x)\ge0$ for all real $x$.

So, $f_+$ and $f_-$ are increasing differentiable convex functions that agree exactly on the countable set $\pi\mathbb Z$.

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Moreover, one can modify this example to make the derivatives of $f_+$ and $f_-$ agree on the set $\pi\mathbb Z$ as well. This can be done by the formula $$f_\pm(x):=e^x(1+c\sin x+\tfrac12\,(c\pm c)\sin^2x )$$ for some $c\in(0,1/4]$ and all real $x$.