Suppose an analytic function $f$ agrees with $\tan x$, $0 \leq x \leq 1$. Could $f$ be entire?
Since $f$ and $\sin z/\cos z$ agree at a set of points and both are analytic in an open neighborhood of $(0,1)$, $f(z)=\sin z/\cos z$ in that open neighborhood. Now the problem for $f$ being entire is that $\sin z/\cos z$ is not differentiable at $z=(2n-1)\pi/2$, which aren't in $(0,1)$. I'm not sure if this is problem or if we can let $f$ be a different analytic function at the points where $\sin z/\cos z$ isn't defined.