Let f(x) be defined for all x>0 and be continuous. Let f(x) satisfies fxy=f(x)−f(y) for all x,y and f(e)= 1. Then

f(x) is continuous for all x>0 and fxy=f(x)−f(y) Also, f(e)=1 Therefore, clearly, f(x)=loge⁡x satisfies all these properties.  Thus, f(x)=loge⁡x, which is an unbounded function.