What are the reasons of these contradictory results?

Question

Aufgabe:

What are the reasons of these contradictory results?

Problem/Ansatz:

This is a short article proving the continuity of a kind of Theta Functions. However, we could prove in a previous article that the same function is not continuous in any point of its curve. What causes this contradiction ? Can we blame the formula of Poisson which is used in Riemann Zeta Function? Here is the link:
https://www.researchgate.net/publication/377271280_A_mathematical_letter_contradicting_the_discontinuity_of_the_function_of_irrationals