Tessellate S.T.E.M.S - Mathematics - College - Set 3 - Problem 4

Let $U$ denote the unit cell $[0,1] \times [0,1] \subset \mathbb{R}^2$ .

Let $\{f_i\}_{i \in \mathbb{N}}$ be a sequence of continuous functions $f_i : \mathbb{R}^2 \to \mathbb{R}$ such that

• $\displaystyle T_i = \oint_{U} f_i \quad$ is finite $\forall \space i$ .
• $f_i \left( \dfrac{a}{m} \right) = 0 \space \space$ for integers $a, m$ such that $0< \left|m\right| < i$

Which of the following is correct about the sequence $\{ T_i\}_{i \in \mathbb{N}}$ ?

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This problem is a part of Tessellate S.T.E.M.S.