Do holomorphic functions map boundaries to boundaries?

I should first say that I am new to Real/Complex Analysis.

Say we have some holomorphic function f : C -> C, and we want to find the image under f of some subset U of C, which has boundary ∂U. Can we say that the image under f of the boundary is the boundary of the image under f of U? i.e. is f(∂U) the boundary of f(U)?

As an example, lets take f(z) = (z-1)/(z+1), and U to be the set of all complex numbers with real part greater than zero (so ∂U is the imaginary axis). Then f(∂U) is the circle of radius 1 centred at the origin, and we can check that f(U) is the set of all complex numbers with magnitude less than 1. So we have that f(∂U) is the boundary for f(U).

I have encountered several examples like this where it seems to hold. Is it true in general?