A function $f$ is known to be continuous and its (simple) functional form is known only for rational numbers. Does it imply that the same functional form is valid for any real number?
For example, I am trying to find the relationship between two magnitudes, I know that it is represented by a continuous function, and I proved that $f(x)=x^2$ but the proof is only valid for $x$ rational. Does this also imply that the relationship is $f(x)=x^2$ for any real value of $x$?