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TIFR Mathematics 2023 | Part B | Question: 2

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Answer whether the following statements are True or False.

Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions such that $f^{2}+g^{2}$ is uniformly continuous. Then at least one of the two functions $f$ and $g$ is uniformly continuous.

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False.

sin(eˣ) and cos(eˣ) are continuous on ℝ, and are not uniformly continuous

 sin²(eˣ)+cos²(eˣ) = 1 is uniformly continuous.
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