Interval analysis is applied to the fixed-point problem x=ϕ(x) for continuous ϕ:S→S, where the space S is constructed from Cartesian products of the set R of real numbers, with componentwise definitions of arithmetic operations, ordering, and the product topology. With the aid of an interval inclusion φ:IS → IS in the interval space IS corresponding to S, interval iteration is used to establish the existence or nonexistence of a fixed point x*of ϕ in the initial interval X0. Each step of the interval iteration provides lower and upper bounds for fixed points of ϕ in the initial interval, from which approximate values and guaranteed error bounds can be obtained directly. In addition to interval iteration, operator equation and dissection methods are considered briefly.