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\begin{center}
  {\Large 2021-22 MAT257 Term Test 2 Rejects}
\end{center}

The following questions were a part of a question pool for the 2021-22 MAT257 Term Test 2, but at the end, they were not included.

\begin{enumerate}

\item Prove that the set of irrational numbers is not of measure-0.

\item Prove that the collection of all finite sequences of rational numbers is countable.

\item Given a set $A$, an ``accumulation point'' for $A$ is a point $x$ such that every open neighbourhood of $x$ contains infinitely many elements of $A$. Show that if $A$ is bounded and has finitely many accumulation points, then $A$ is of content 0.

\item Prove that every closed set is the intersection of countably many open sets.

\item Give an example of two functions that differ only on a bounded set of measure 0, yet such that one is integrable and the other is not.

\item Use Fubini's Theorem to compute the volume of the set $\{x\in\bbR^5\colon 0\leq x_1\leq x_2\leq x_3 \leq x_4 \leq x_5 \leq 1\}$.

\item Show that there is a smooth function on $\bbR^3$ whose support is precisely the cube $[-1,1]^3$.

\item Find an example of a continuous function on $\bbR$ for which there is a constant $M$ such that $\int_If\leq M$ for every internal $I\subset\bbR$, but yet such that $f$ is not integrable (NT).

\item We've shown in class that $\int_{\bbR^n}e^{-|x|^2/2}dx=(2\pi)^{n/2}$. Let $\lambda$ be a positive real number. Compute $\int_{\bbR^n}e^{-\lambda|x|^2/2}dx$.

\end{enumerate}

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