Like the discrete uniform RV, the continuous uniform RV U gives equal weight to all elements in the space of outcomes, which must be an interval [a,b]. This means that the density function of U must be constant over the range of possible outcomes: f(u)=c du for all u . The value of the constant is determined by the requirement that the density function must integrate to 1. Since the integral of a constant over the interval a to b is just c(b-a), it follows that the constant is c=1/(b-a).
S2: Three Uniform Densities and Limits
S3: CDF of Uniforms