Let $s=\sigma+it$, then the approximate functional equation for Riemann zeta function can be given by $$\zeta(s) = \sum_{n\leqslant \sqrt{|t|/(2\pi)}}\frac{1}{n^s} \ + \chi(s) \ \sum_{n\leqslant \sqrt{|t|/(2\pi)}}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1})\tag{1}$$ where $\chi(s)=\frac{\pi^{-(1-s)/2}\Gamma((1-s)/2)}{\pi^{-s/2}\Gamma(s/2)}$.
Question Because the number of summation terms jumped by one when $t$ varies from $8\pi+0^-$ $8\pi+0^+$, is $\zeta(\sigma+it)$ still continuous at $t=8\pi$? There must be a proof somewhere.