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evinda

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I want to find two convex functions $f,g: \mathbb{R} \to \mathbb{R}$ such that $f(x)=g(x)$ iff $x$ is an integer.I have thought of the following two functions $f(x)=e^x$, $g(x)=1$.

Then at the $\Rightarrow$ direction, we would have $f(x)=g(x) \Rightarrow e^x=1 \Rightarrow x=0 \in \mathbb{Z}$.

Right?

At the other direction, we cannot pick $0$, we have to pick an arbitrary integer. Right? If so, then it does not hold that $f(x)=g(x)$...

So do we have to pick other $f,g$ ? (Thinking)