Show if each of these functions fx RR is Borel measurable E

Show if each of these functions f(x) : RR, is Borel measurable? Explaine please!

1/ f(x) = sinx

2/ f(x) = 1/sinx if x not equals k,k Z, and 1 if x = k,k Z

3/ f(x) = sin(1/x) if x not equals 0, and 1/2 if x = 0

4/ f(x) = x if x Q, 0 otherwise

Solution

here

f(X)=sinx and

f(x)=x if if x Q, 0 otherwise

are both contineous functions

and f(x)=x is also countable as x Q, (and every countable set is borel measurable with measure 0)

therefore these are borel measurable

(because every contineous function is measurable hence borel measurable)

and f(x) = 1/sinx if x not equals k,k Z, and 1 if x = k,k Z

             f(x) = sin(1/x) if x not equals 0, and 1/2 if x = 0 are not contineous at x = k,k Z

therefore these functions have infinite no. of points of discotinuty

but both of these functions gives countable sets when we put integers in it .{ because here k Z(set of integres) and set of integers are countable}

therefore these are borel measurable

(because every countable set is measurable hence borel measurable)

note: if function is not contineous then it may be measurable