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Measurable Functions

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Prove directly from Proposition 17.2 that each of the following function is measurable on [0,1].
a) f(x)=3 for all x in [0,1]
b) f(x) =x for all x in [0,1]
c) f(x)=1 if x in (0,1], f(x)=0 if x=0.

I have also included the proposition in the attachment
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The expert examines measurable functions.

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Take A=[0,1]. Then:
a) Consider two cases: c<3 and c>=3. If c<3, then the set {x in A: f(x)<c} is empty, and so is measurable.
If c>=3, then the set {x in A: f(x) < c} = [0,1] and so is ...

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