Let f:R->R satisfy f(x+y)=f(x)f(y) for all x,y in R. Suppose f has a limit at 0, prove that f has a limit at all points. f could be either an exponential function which base is non-zero, or identically 0 or 1. Of course, this can't serve to prove anything unless I can prove that f cannot be anything else. f(x)=f(0+x)=f(0)f(x). If f isn't identically 0, then f(0)=1. However I fail to see how the limit of f at 0 can help me. Any pointers will be appreciated.