Question

Consider the function f(x) = {0, x = 0 and 1 - x, 0 <= x <= 1}. Which of the following statements is false?

a. f is differentiable on (0, 1).
b. f(0) = f(1)
c. f is continuous on [0,1]
d. The derivative of f is never equal to zero on the interval (0,1).

B and D are true, and it appears continuous, so A is my answer.

Answer 1

A is true, since (0,1) is an open interval. Pick any nonzero x, and the limit from both sides is just 1-x.

B and D are also true.

C is false, since f(0) = 0 and f(x)->1 as x->0+

f is continuous on (0,1]

There is a problem with the definition. It should be

f(x) =
0 if x=0
1-x if 0 < x <= 1