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I describe here a function $g$ (based on Thomae's function) that is nowhere locally bounded. In particular the image of any interval $(a,b)$ under $g$ is an unbounded segment of the integers $\mathbb{N}$.

A partial plot of $g$

Is it possible to find a function such that the image of any interval $(a,b)$ would be the full set of the reals?

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I guess you are looking for the Conway base 13 function which is an example for the more general class of strongly Darboux functions. These share the property of mapping every open interval of the reals to the whole line.

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