Final answer:
The −NMe₂ group in N,N-dimethylaniline functions as both a π-donor, because of the unbound pair of electrons on the nitrogen, and as a sigma-donor, through its polar C-N bonds and interactions with the two methyl groups.
Step-by-step explanation:
The −NMe₂ group in N,N-dimethylaniline is indeed capable of exhibiting both π-donor and sigma-donor characteristics. This group contains an unbound pair of electrons on the nitrogen atom, which allows it to act as a π-donor. Due to nitrogen being more electronegative than the carbon to which it is bonded, the C-N bonds are polar, contributing to the sigma-donor properties.
The lone pair of electrons on the nitrogen can be donated to a proton, rendering the nitrogen positively charged and more reactive. Moreover, the two methyl groups attached to the nitrogen in the −NMe₂ group further contribute to its electron-donating capacity through the sigma bonds.