Piecewise-constant function with infinitely many pieces
That $F$ should really be something like $F_n$, defining a family of functions and then your total $F$ would really be the union of the $\{F_n\}$ over the domains where they are non-zero.
Then unless I am much mistaken each of those domains will have length $2$ and so we can define your family of functions as you had above and then have a dispatcher function to the appropriate function be your union function, using Quotient
So in total it will look like:
fxn[x_, n_] :=
Piecewise[
{
{x - n, 2 n <= x <= 2 n + 1},
{n + 1, 2 n + 1 <= x <= 2 n + 2}
}
];
fxn[x_] :=
fxn[x, Quotient[x, 2]]
And we'll confirm that I have this right:
With[{maxN = 10},
{
Plot[
Evaluate@Table[fxn[x, n], {n , 0, maxN}], {x, 0, 2*maxN + 2}],
Plot[fxn[x], {x, 0, 2*maxN + 2}]
}
]
"... I need it only as a function of x"
ClearAll[f]
f[x_] := Min[x - Quotient[x, 2], 1 + Quotient[x, 2]]
Plot[f[x], {x, 0, 10}]
Plot[f[x], {x, -5, 10}]
Plot[f[x], {x, 0, 50}]
One more way is as follows.
f[x_] := Sum[ Piecewise[{{x - n, 2 n <= x <= 2 n + 1}, {n + 1,
2 n + 1 <= x <= 2 n + 2}}], {n, -Infinity, Infinity}];
Plot[f[x], {x, -4, 5}]