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



Intersecting points of increasing convex functions
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 $x$.


