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 ( x 0 ) − g ( x 0 ) | , 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.