Continuity of Functions on Metric Spaces Let x0 be a particular point in a metri

Question

Continuity of Functions on Metric Spaces Let x0 be a particular point in a metric space (X, d). Show that the function f on X given by f(x) = d(x,x0) is continuous.

Explanation / Answer

Suppose that f were not continuous at p. Then for some > 0 we cannot find any choice of to satisfy the continuity condition. In particular, = 1 will not work. Hence for some point x1 we have dX(x1 , p) < 1 but dY(f(x1), f(p)) . Similarly = 1/2 will not work and so for some point x2 we have dX(x2 , p) < 1/2 but dY(f(x2), f(p)) , ... Continue like this to get a sequence (x1 , x2 , x3 , ...) with dX(xn , p) < 1/n but dY(f(xn), f(p)) for each n. But since (xn) has been constructed so that (xn) p this contradicts the condition given in the theorem. hence it is continous

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

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