We can solve **M first.

Let 3x+ 1=n+a (n is an integer, 0 ≤ a

f(x)-2x = n+0.5-2(n- 1+a)/3 = 0

So 3n+ 1.5-2n+2-2a=0, that is, n=2a-3.5.

Because 0 ≤ 2a

So-3.5 ≤ n

So n=-3 or -2, corresponding to a=0.25 or 0.75.

So x=- 1.25 or -0.75 and then look at g(x)-2x=0.

-1≤x≤0, g (x) = 2 (-x)-1≥1= 0, x=0, equals sign, 2x≤0.

So g(x)=2x can only be obtained when x=0.

When 0

So only when x= 1 can we get g(x)=2x.

When 1

So g(x)=2x can only be obtained when x=2.

The same is true in 2.

So the solution of g(x) is x=0, 1, 2.

So the sum of all the elements is:

- 1.25-0.75+0+ 1+2= 1