If an analytic function is constant along the imaginary axis, is it constant on it's entire domain?

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I can't seem to prove this but I also cannot find a counter example. I am dealing with the idea of a function that is constant on the imaginary axis and is defined through an unbound domain. Is it a constant function?If so, why? And if it isn't, can you show me such function?

I would like to note that this is not a function with ImF being constant, but F is constant when fed imaginary numbers.

Thank aplenty.