Let S R and suppose f is a function defined on S. The function f is called Lipsc

Question

Let S R and suppose f is a function defined on S. The function f is called Lipschitz if there exists a bound M > 0 so that |f (x) -f(y)/x - y| lessthaorequalto M for all x, y element S. Geometrically speaking, a function is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f Show that if f is Lipschitz, then f is uniformly continuous on S Is the converse statement true? In other words, are all uniformly continuous functions necessarily Lipschitz?

Explanation / Answer

Given if a function is Lipschitz function;

that means |f(x)-f(y)/(x-y)|<=M;

for all x,y belong to S;

let x = y+h;

we can say that

|f(y+h)-f(y)/h|<=M

for h-> 0; |f'(y)|<=M;

hence f'(y) exists or f is differentiable function for all y belong to S;

so f is continuous function;

B) No not all uniformly continuous function are not Lipshitz function;

as it is not neccesarily true that function's slope will be bounded;

eg. let f be a circle;

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.