BackRemovable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.
If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.
Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.
The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).
Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 3x − 1 = 0
Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 15x + 1 = 0 (three roots)
At what points are the functions in Exercises 13–30 continuous?
y = cos (x) / x
At what points are the functions in Exercises 13–30 continuous?
y = √(2x + 3)
At what points are the functions in Exercises 13–30 continuous?
g(x) = { (x² − x – 6)/(x – 3), x ≠ 3
5, x = 3
Is there a value of c that will make
f(x) = { (sin²(3x)) / x², x ≠ 0
c, x = 0
continuous at x = 0? Give reasons for your answer.
Limits and Continuity
Graph the function
1 , x ≤ ―1
―x , ―1 < x < 0
ƒ(x) = { 1 , x = 0 ,
―x , 0 < x < 1
1 , x ≥ 1
Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.