So I gave it a go, and I just want to make sure my argument is convincing: If f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞(notice the closed brackets, I wanted to leave the option that c can be infinity, is this how I should write it? if not, how?), because if this wasn't true, then lim x->cf(x)=A (A is finite) for all c in [0,∞], but then f would be bounded (is this convincing?). Now suppose c is in [0,∞) (open bracket, so c isn't infinity), then lim x->cf(x) = ±∞, but since c is finite and f is continuous, then by the definition of continuity f is defined at c, but then f(c)= ±∞, but ±∞ isn't even in the reals so then f wouldn't be undefined at c, so it wouldn't be continuous. So c=∞, and lim x->∞f(x) = ±∞. Thoughts?