Question 1020688
a. S.  
Take for example the continuous function {{{f(x) = system(2x-1 for x>=0, -1 for x<0)}}}, 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 {{{f(x) = system(-x-2 for x<=0, 3x-2 for x>0)}}}, where f(-1) = -1, f(1) = 1, but f(0) = -2. On the other hand, f(x) = x and {{{f(x) = x^3}}} satisfy the given condition.

d.  A.
This is the intermediate value theorem.

e.  S.
{{{f(x) = x^3}}} satisfies the given condition, but {{{f(x) = x - x^3}}} 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.