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Problem 32: (Periodic Functions And Practice With Quantifiers)

Consider the following definition:

We say that a function $y=f(x)$ is periodic over the reals if there exists a positive real number $k$ such that $f(x+k)=f(x)$ for every real number $x$.
  1. Rewrite the definition above using the symbols $\exists$ and $\forall$.
  2. Prove that $y=\sin(x)$ is periodic over the reals.
  3. Finish the following: "We say that a function $y=f(x)$ is not periodic over the reals if ..."
  4. Prove that $y=x^2$ is not periodic over the reals.

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