AL Pure Maths - function II

(a)

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(b i) Obviously f is NOT injective since f(x) = 0 for all 0 <= x < 1.
(ii) Analysing for:
1 <= x < 2, f(x) = sin πx, covering the range between 0 and -1.
2 <= x < 3, f(x) = 2 sin πx, covering the range between 0 and 2.
3 <= x < 4, f(x) = 3 sin πx, covering the range between 0 and -3.
4 <= x < 5, f(x) = 4 sin πx, covering the range between 0 and 4.
Contiunuing with increasing x, it is found that f(x) covers all real values and hence it is surjective.
(c i) For x not being an integer:
f(x) = N sin πx where N = [x]
And so it is differentiable.
For x being an integer, say x = M
f+'(x) = d(M sin πx)/dx with x = M
= πM cos Mπ
f-'(x) = d[(M - 1) sin πx]/dx with x = M (since x < M, [x] = M - 1)
= π(M - 1) cos Mπ
So the left and right hand derivatives are unequal, meaning that when x is an integer, f is NOT differentiable.