Is this function just zero?

Question

I asked a question like this a few minutes, but am about ready to strike the problem out as errata for the book. The problem defines a function $g(x)=f(x-a)g(b-x)$. The function has bound $|f|<1$, so I believe the equality, $g(x)=f(x-a)g(b-x)=g(x)=f(x-a)f(b-a-x)g(x)$ shows that $g(x)=0$. Since the author claims that this is a bump functions, positive on $(a, b)$, I think he meant $g(x)=f(x-a)f(b-x)$. Any opinions? The book is Differential Topology by Alan Pollock. Sec 1, Excersise 18 FYI.