AP Calculus Problem. Is the above function continuous, differentiable? What is the derivative

The amount of federal income tax a single person with a taxable income of 77,100 or less must pay is listed below
10% of taxable income up to 7,825
15% of taxable income more than 7,825 up to 31,850
25% of taxable income more than 31,850 to 77,100

a. is the above function continuous? Is it differentiable? Write set notation
b. Write and graph the derivative of the function
c. Explain the meaning of the derivative in context.

Suppose that a person has a taxable income of x. Then, the taxation function is:
f(x) = { 0.10x for 0 <= x <= 7825
. . . . .{ 0.15x for 7825 < x <= 31850
. . . . .{ 0.25x for 31850 < x <= 77100.

(a) 0.10x, 0.15x, and 0.25x are all polynomials and polynomials are continuous, so the only place where f(x) could be discontinuous is when the piece-wise function "skips" at x = 7825, x = 31850, and x = 77100. You can show that:
(i) lim (x-->7825-) f(x) ≠ lim (x-->7825+) f(x)
(ii) lim (x-->31850-) f(x) ≠ lim (x-->31850+) f(x)
(iii) lim (x-->77100-) f(x) ≠ lim (x-->77100+) f(x),

so f(x) is not continuous at x = 7825, 31850, and 77100. In set-builder notation:
D = {x | x ≠ 7825, 31850, and 77100}.

(b) By differentiating f(x) piece-by-piece:
f'(x) = { 0.10 for 0 <= x <= 7825
. . . . .{ 0.15 for 7825 < x <= 31850
. . . . .{ 0.25 for 31850 < x <= 77100.

You can show that f'(x) is continuous for all x ≠ 7825, 31850, and 77100. So, f'(x) is differentiable for:
D = {x | x ≠ 7825, 31850, and 77100}.

(c) Note, that for every dollar one makes, one are taxed either $0.10 (if his income is equal to or less than $7,825), $0.15 (if his income is more than $7,825 but equal to or less than $31,850), or $0.25 (if his income is more than $31,850 but equal to or less than $77,100). This is what the derivative means.

I hope this helps!