Let f :[a,b] ---->R be a continuous function and likewise let g:[c,d] -----> R be another continuous function. Where R represents the set of real numbers, [a,b] and [c,d] are subsets of R

Next we define a function P : [a,b] x [c,d] -----> R by P(x,y)=f(x)g(y). Is it possible to "construct" the continuous functions f and g such that (x_{1 , }y_{1}) # (x_{2}_{ , }y_{2}) implies P(x_{1 }, y_{1}) # P(x_{2}, y_{2})

Note that f and g have to be continuous functions defined on their respective domains, namely [a,b] and [c,d]

I have "struggled" with this issue for months and cannot get a concert answer

Please advise

Thanks