SOLUTION: 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

SOLUTION: 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 som

Question 1020730: 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 richard1234(7193) (Show Source): You can put this solution on YOUR website!
a) Sometimes true (f(0) need not be 0)
b) Always true, by continuity/intermediate value theorem
c) Sometimes true, f could jump outside the interval [-1,1]
d) Always true, also holds from intermediate value theorem
e) Sometimes true, we don't know anything about f
f) Never true, since f is discontinuous at x = 0
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