Real Analysis question about uniform continuity.

I know that a constant function is uniformly continuous.

We have a function f: [0,+inf] - > R

Say f(x) = 1 for x in [0,1] and f(x) = 2 for x in [1,+inf)

Is f uniformly continuous on [0,+inf) ?

I think that if instead f(x) = 1 for x in [0,1) (not including 1) and f(x) = 2 for x in [1,+inf), then it would not be continuous at 1 so it cannot be uniformly continuous on [0,+inf). But since it's [0,1] and [1,+inf), I'm not sure if it is or is not.

Thanks!