Continuity Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Sketch the graph of the function

In exercises (2)-(12) use the function given in exercise (1).

(2) Does exist?

(3) Does the limit, exist?

(4) Does the limit, ?

(5) Does exist?

(6) Does the limit, exist?

(7) Does the limit, exist?

(8) Is defined at

(9) Is continuous at

(10) At what values is continuous?

(11) What value should be assigned to , to make the extended function continuous at

(12) To what new value should be assigned to remove the discontinuity?

(13) What is the largest set on which will be continuous?

(14) What is the largest set on which will be continuous?

(15) What is the largest set on which will be continuous?

(16) What is the largest set on which will be continuous?

(17) Find constants and so that the function will be continuous for all in the domain.

(18) Find constants and so that the function will be continuous for all in the domain.

(19) Find constants and so that the function will be continuous for all in the domain.

(20) Let and . Show that

(21) If a function is not continuous at , but can be made continuous at by being assigned a new value of that point, it is said to have a removable discontinuity at Which of the following functions have a removable discontinuity at ?

    (a) at
    
    (b) at
    
    (c) at     

(22) Prove that the function must have at least one real root.

(23) Prove that the function must have at least one real root.

(24) Find a function(s) with the following properties.

    (a) Find functions and such that is discontinuous at but is continuous there.

    (b) Give an example of a function defined for all real numbers that is continuous at only one point.

• print this page
• pages linking here
• related pages