The function f (x) = cx is convex, for every x (strict convexity is not required for the Legendre transformation to be well defined). Clearly x * x − f (x) = (x * − c) x is never bounded from above as a function of x, unless x * − c = 0. Hence f * is defined on I * = { c } and f *(c) = 0. (The definition of the Legendre transform requires the existence of the supremum, that requires upper bounds.)