Please solve this question Please justify your answer Let f

Please solve this question. Please justify your answer.

Let f: R rightarrow R. Suppose that | f(x) - f(y)| lessthanorequalto M middot |x - y|, for all x, y R, where M greaterthanorequalto 0 is fixed. Show that f is continuous. (A function f with this property is actually called Lipschitz continuous; so, this exercise shows that any Lipschitz continuous function is continuous.)

Solution

SOLUTION) SINCE THE GIVEN FUNCTION SATISFIES THE LIPCHITZS CONDITION THEN BY A THEOREM

f(x) IS UNIFORMLY CONTINUOUS BECAUSE EVERY FUNCTION WHICH SATISFIES THE LIPCHITZ CONDITION IS UNIFORMLY CONTINOUS AND EVERY UNIFORMLY CONTINUOUS FUNCTION IS CONTINUOUS.THEREFORE THE GIVEN FUNCTION IS CONTINUOUS.