must a bounded continuous function on R be uniformly continuous?

I know that if a function is continuous on a closed and bdd set then its uniformly continuous, but this says nothing about the set, just the function. Would this be T or False? How would I go about proviing it?

Also, if f and g are uniformly continuous maps of R to R, must the product f*g be uniformly continuous? What if f and g are bounded.

I think this one is true...since you're just multiplying two continuous functions together...but I don't know how to prove this either

Thanks.