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### Calculus - Functions, Limits, Continuity Problem Set III - Examples, Problems related to Continuity, Intermediate Value Theorems :

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## Calculus - Functions, Limits and Continuity - Problem Set III - Outline of Contents:

Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

### Here's a quick look at the kind of problems we will go through, in this tutorial - Problems related to Continuity, intermediate value theorem :

For what values of x is the following function continuous?
(x−1) /  √x - 1 if  x>1
5 − 3x, if  −2 ≤ x ≤ 1
6/x−4 ,  if  x < −2

Determine the values of the constant A so that the following
function is continuous for all values of x.
f(x) =
A2 x − A, if x ≥ 3
4,
if x < 3

Determine the value of constants A and B so that the following
function is continuous for all values of x.
Ax − B, if
x ≤ −1
2x2 + 3Ax + B, if −1 < x ≤ 1
4,
if
x>1

Show that the following function is continuous for all values of x.
f(x) =
e−1/x2 , if x != 0
0  if x = 0

Show that p(x) = 2x3 − 5x2 − 10x + 5 has a root somewhere in
the interval [−1, 2].

Let f be a function on [0, 1]. Show that if −1 ≤ f (x) ≤ 1 for all x ∈ [0, 1] then there is c ∈ [0, 1] such that [f (c)]2 = c.