The short answer is yes, a theoretical, ideal, resonant dipole has no reactance. This statement break down at the ends of the dipole because there cannot be any current at and beyond the wire end.

In practice, a dipole has distributed resistance, capacitance and inductance due to material properties, physical dimensions and the environment (e.g. height above ground).

The link you provided actually does have the answer to your question if you study the graph carefully. I.e. reactance is close to zero up to approx 1/8 of a WL from the center. This can be used to T-match a dipole to a 300Ω or 450Ω balanced feed line.

Note that if you asymmetrically feed a 0.5 WL antenna (i.e. not in the center) then you no longer have a resonant dipole and on the longer section a phase reversal will occur.

More information: dipole feed matching and with a bit more theory: Practical Dipole/Monopole Geometries. Matching Techniques for Dipole/Monopole Feeds