Question 1020688: Let f be a continuous function such that f(-1) = -1 and f(1) = 1.
Classify each of the following statements as:
A - ALWAYS TRUE
N - NEVER TRUE
S - SOMETIMES TRUE - true in some cases, false in others
Justify each. Explain.
a. f(0) = 0
b. For some x with -1 <= x <= 1, f(x) = 0
c. For all x with -1 <= x < 1, -1 <= f(x) <=1
d. Given any y in [-1,1], then y = f(x) for some x in [-1,1].
e. If x < -1 or x >1, then f(x) < -1 or f(x) > 1
f . f(x) = -1 for x < 0 and f(x) = 1 for x > 0
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! a. S.
Take for example the continuous function  , where f(0) = -1. On the other hand, f(x) = x gives f(0) = 0
b. A.
Since f(1) = 1 and f(-1) = -1 and f(x) is continuous, then by the intermediate value theorem, there is a c in [-1, 1] such that f(c) = 0.
c. S.
Take for example the continuous function  , where f(-1) = -1, f(1) = 1, but f(0) = -2. On the other hand, f(x) = x and  satisfy the given condition.
d. A.
This is the intermediate value theorem.
e. S.
satisfies the given condition, but  is the complete opposite.
f. N.
In fact if it were true, the function would be discontinuous at x = 0, contradicting the hypothesis that f(x) is continuous.
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