# Continuity of a function

there is one problem. the problem is related with contuinity of afunction and i tried like as shown below.so if any one who is intersted to help me i like ..
the problem is
prove that if f(a+b)=f(a)f(b) for all a and b ,then f is cntiniuous at every real number.here there is given information that is the domain of f is the set of all real number and f is continuous at 0.
I tried to approache this problem like this
since f is continuous at 0, f(0) must defined and it must be equal to the value of limf(x) =f(0)
x-->0
and a and b are real numbers then limf(a+b)=limf(a).limf(b)
x--->0 x-->0 x-->0
=f(a).f(b) (since f(a) and f(b) are constnt fu.)
this implies f is continuous at every real number.

$$\lim_{x \to a} f(x) = \lim_{x \to a} f( (x - a) + a )$$.