Final answer:
The function f is shown to be continuous by using the properties of the continuous function g and the provided inequality. By setting up the limit as x approaches any point a, and applying the given conditions, it's proved that the limit of f(x) equals f(a), satisfying the definition of continuity.
Step-by-step explanation:
Proving Continuity of Function f(x)
To show that the function f is continuous, we'll make use of the given properties of g and the inequality |f(x)−g(y)|≤g(x−y). The function g is known to be continuous and satisfies the condition g(0)=0. Now, to show that f is continuous at any point a in R, we consider the limit of f as x approaches a and use the given inequality.
Given ∀x,y ∈ R, |f(x)−g(y)|≤g(x−y), set y = a and take the limit as x approaches a. Utilizing the continuity of g, we have limx→ag(x−a) = g(0) because x−a approaches 0 as x approaches a. Because g(0)=0, the right-hand side of our inequality approaches 0.
Thus, we have limx→a|f(x)−g(a)| = 0, which implies f(x) approaches g(a) as x approaches a. Since this must hold for all a in R and g is continuous, it follows that f(a) = g(a), and therefore the limit of f(x) as x approaches a equals f(a), which is the definition of continuity of f at a. This completeness of the argument shows that f is continuous over R.