ThereFor any given exponential function <math>f</math>, there are infinitely many functions whose self-composition is the same exponential function as each other<math>f</math>. In particular, for every <math>A</math> in the [[open interval]] <math>(0,1)</math> and for every [[continuous function|continuous]] [[Monotonic function|strictly increasing]] function <math>g</math> from <math>[0,A]</math> [[surjective function|onto]] <math>[A,1]</math>, there is an extension of this function to a continuous strictly increasing function <math>f</math> on the real numbers such that {{nowrap|<math>f\bigl(f(x)\bigr)=\exp x</math>.{{r|croneu}}}} The function <math>f</math> is the unique solution to the [[functional equation]]
