Answer Happy

• A function f(x) is said to be piecewise continuous if it is continuous except for a finite set of discontinuity points of the first kind. f(x) defined on an interval I is said to be a Lipschitz function with constant L> 0 provided that |f(x) = f(y)| ≤ L\x-y| for all x and y in I. Clearly any Lipschitz function is uniformly continuous on I. The least constant L with which f satisfies the above Lipschitz condition is said to be the Lipschitz constant of f. Question: 3.1 Suppose that fe C(R) and that f(x + 1)-f(x) converges to 0 as x→ ∞o. Then show that f(x) X also converges to 0 as x→∞o. Suppose further that f(x+y)-f(x) converges to 0 as x→∞o for an arbitrary fixed y. Show then that this convergence is uniform on compact sets in R.