g is a function on R to R which is not identically zero.
if g(x+y) = g(x)g(y) for x,y in R
if g is continuous at every point of R show that g(x)>0 for all x in R
This problem belongs in the calculus subforum.
Suppose g(0) < 0. Then there exists a δ > 0 such that |x| < δ implies g(x) < 0. Consider any x with |x| < δ and g(x) = g(0 + x) to derive a contradiction.
Similarly, if g(0) > 0, then there exists a δ > 0 such that |x| <= δ implies g(x) > 0. Also, g(x + δ) has the same sign as g(x).