Long story short: fourier series is taken in two subjects (for now).
One doc says that the dc component is 0 if the function is odd.
The other says that odd and even has no effect on the dc component; it is the symmetry to x-axis that matters.
Is there a function that is odd , has a dc component and periodic?
On
If by DC component you mean the overall integral of the function (i.e. the amount of the curve above the x axis minus the amount below it) then both are correct in that an odd function is also symmetrical about the y axis. As pointed out, no function is symmetrical over the x axis except $f(x)=0$ but what I assume you meant was it is the same for the positive x axis as it is in the negative x axis.
An odd function is not necessarily periodic, but will never have a dc component.
Also this is a mathematics forum not a physics forum... dc component is not a mathematical term.
The dc component of $x\mapsto f(-x)$ is the same as for $f$. The dc componet of $x\mapsto -f(x)$ is the negative of the dc component of $f$. Hence for odd $f$, the dc component equals its own negative and must be zero.
Also not that no function (except constant zero) is symmetric to the $x$-axis.