Buffer solutions have constant acidity and alkalinity because

If small amount of an acid or alkali is added to a buffer solution, it converts them into unionised acid or base. Thus, its $\mathrm{pH}$ remains unaffected or in other words its acidity/alkalinity remains constant. e.g., $$ \begin{aligned} & \mathrm{H}_3 \mathrm{O}^{+}+A^{-} \rightleftharpoons \mathrm{H}_2 \mathrm{O}+\mathrm{H} A \\ & { }^{-} \mathrm{OH}+\mathrm{H} A \longrightarrow \mathrm{H}_2 \mathrm{O}+A^{-} \end{aligned} $$ If acid is added, it reacts with $A^{-}$to form undissociated HA. Similarly, if base/alkali is added, $\mathrm{OH}^{-}$combines with $\mathrm{H} A$ to give $\mathrm{H}_2 \mathrm{O}$ and $A^{-}$and thus, maintains the acidity/ alkalinity of buffer solution.