Suppose that f has a domain which contains I = {x : a < x< b}, and suppose that

Question

Suppose that f has a domain which contains I = {x : a < x< b}, and suppose that f(x) --> L as s --> b-.Prove the following: for each > 0 there is a >0 such that |f(x) - f(y)| < for all x, y with b - < x < b, b- < y < b. *** PLEASE SHOW ALL WORK!! Suppose that f has a domain which contains I = {x : a < x< b}, and suppose that f(x) --> L as s --> b-.Prove the following: for each > 0 there is a >0 such that |f(x) - f(y)| < for all x, y with b - < x < b, b- < y < b. *** PLEASE SHOW ALL WORK!!

Explanation / Answer

Then if x and y are any two points such that b - d < x <b and b - d < y < b, then we have
| f(x) - L | < e/2 and |f(y) - L| < e/2 and hence addingthese two gives us
|f(x) - L | + |f(y) - L| < e. And now we note that by thetriangle inequality
|f(x) - f(y)| = |f(x) - L + L - f(y)| <= |f(x) - L| + | L -f(y)| = |f(x) - L| + |f(y) - L| < e which gives us theinequality we need.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

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