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.

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

To determine which statement is false, let's analyze each option one by one:

a. f is differentiable on (0, 1).
To check if a function is differentiable at a point, we need to ensure that the function is continuous at that point and that the left-hand and right-hand derivatives exist and are equal.
In this case, f is not differentiable at x = 0 because it has a sharp corner, and the left-hand and right-hand derivatives would be different. Therefore, statement a is true.

b. f(0) = f(1)
To verify if f(0) is equal to f(1), we need to substitute the corresponding values into the function.
f(0) = 0, and f(1) = 1 - 1 = 0. Therefore, f(0) = f(1). Hence, statement b is true.

c. f is continuous on [0,1]
To check continuity, we need to ensure that the function is defined for all x-values in the interval [0,1], and the left-hand and right-hand limits match the function value at the endpoints.
In this case, f is defined and continuous on [0,1] because both f(0) and f(1) exist (as established in statement b). Thus, statement c is true.

d. The derivative of f is never equal to zero on the interval (0,1).
To find the derivative of f, we differentiate each piece of the function separately. Since f(x) = 0 for x = 0, the derivative is also zero at that point. However, for 0 < x <= 1, the slope of the line is constantly -1, which means the derivative is always -1 for x in (0,1). So, statement d is false.

Hence, the false statement is d.