The Lipschitz metric for real-valued continuous functions

For a continuous real function f defined on a metric space X, let α(f) denote its minimal Lipschitz constant if f is Lipschitz and put α(f)=∞ otherwise. We study the extended real-valued metric on the continuous real functions defined by d(f,g)=max{|f(x0)−g(x0)|,α(f−g)}. When X=[a,b] this metric provides new insight into a classical result regarding the derivative of a limit of a sequence of real-valued functions defined on the interval.