Question: Show that if I := [a, b] and f : I R is increasing on I, then f is continuous at a if and

Show that if I := [a, b] and f : I → R is increasing on I, then f is continuous at a if and only if f (a) = inf{ f(x) : x ∈ (a, b]}.

Step by Step Solution

★★★★★

3.36 Rating (180 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

If L inffx x a b and 0 then there exists x a b with L fx L Since ... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Document Format (1 attachment)

Word file Icon

829-C-F-M (409).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Q:

Let f : R R be defined by f(x) = x2 for x R. (a) Show that the inverse image f-1(I) of an open interval I := (a, b) is either an open interval, the union of two open intervals, or empty, depending...

Q:

Show that if one event A is contained in another event B (i.e., A is a subset of B), then P(A) P(B). For general A and B, what does this imply about the relationship among P(A B), P(A) and P(A B)?

Q:

Suppose that a a) Prove that if f is continuous at x0 [a, b] and f(x0) 0, then b) Show that if f is continuous on [a, b], then ba |f(x)dx = 0 if and only if f(x) = 0 for all x [a, b]. c) Does part...

Q:

Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the exam, or if you prefer, create a document...

Q:

7. A function f : D CR -> R is called (sequentially) lower semicontinuous at a point I C D if we have f(x) R is lower semicontinuous in every z ( [a, b], then f attains its minimum value in [a, b]....

Q:

Math 2720-Fall 2017-Practice final exam Instructor: Raphael Clou atre Date: December 8th, 2017 Length: 2 hours and 30 minutes There are 8 problems on the exam. Notes, calculators or cellphones are...

Q:

nx (a) Let f (0, 1) R be given by fn(x) = +1. Show that {f} converges pointwise to a continuous function f, but the convergence is not uniform. (Remark: This shows that pointwise convergence to a...

Q:

Please show all work, step by step, written and explained as neatly as possible. Thank you so much!! Directions Let f : (cub) > R be u uniformly continuous function. Finish the proofof Theorem 3.4.6...

Q:

Please give full explanations. Question 2: (Continuity)-(12 points) (a): Let f : R' -> R be given by r2 In(r; + .;) . (21,12) / (0,0) ($1, 12) = (0,0). Show that: Of is defined for all (21, r2). is...

Q:

Question 2: (Continuity)-(12 points) (a): Let f : R' -> R be given by r2 In(r; + .;) . (21,12) / (0,0) ($1, 12) = (0,0). Show that: Of is defined for all (21, r2). is not continuous at (0, 0). (b):...

Q:

Carlsbad Gas and Electric is planning to purchase a degassing tower for removing CO 2 from acidified saltwater. The supplier quoted a price of $125,000 if the unit is purchased within the next 3...

Q:

Review the following situations and determine the employee's net pay by figuring out what earnings & benefits are subject to Income Tax, Canada / Quebec Pension Plan, Quebec Parental Insurance Plan...

Q:

Interest Income ( LO 2 . 9 ) Rebecca , a single taxpayer, owns a Series I U . S . Savings Bond that increased in value by $ 4 6 during the year. She makes no special election. How much income must...

Q:

Keith is with his prospect, and he is out on a test drive for the new Ford F-150. He is telling the prospect about the safety features of the truck. What stage of the selling process is Keith in? a....

Q:

Multiply. 1. (240)(1)(20) 2. 1357 79 3. 807 127 4. 1234 567 5. 426 110

Q:

Find the area and the perimeter of each rectangle. See Example 6. 1. 2. 3 inches 13 inches 25 centimeters 20 centimeters

Q:

Estimate the products by rounding each factor to the nearest hundred. See Example 9. 1. 982 650 2. 111 999

Q:

You took a 3 0 - year half - million - dollar mortgage at 7 . 5 % per year. What percent of your last payment goes to reduce principal

Q:

I am planning to test whether the new fertilizer which has come in market increases yield of corn or not. For this I take 200 corn saplings, assign 100 randomly to the new fertilizer and 100 to no...

Q:

As stated in your text, "A longer time horizon when investing in a portfolio of stocks and treasury bills and the law of averages is thought to imply that because of its positive risk premium, the...