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Problem 1.

Prove that the function \(f : \left( 0, \infty \right)\to \left[ 0, 1 \right]\) given by \(f(x) = \sin \left( \frac{1}{x} \right)\) is continuous but not uniformly continuous on its domain.

Problem 2.

Let \(f: (a, b) \to \mathbb{R}\) be a continuous function. Prove that the range of \(f\) must be an interval.

Note: You cannot use the extreme value theorem because the domain is not a closed interval!

Problem 3.

Define the function \(f:\mathbb{R}\to \mathbb{R}\) \[f(x) = \begin{cases} 1 & x \in \mathbb{Q}, \\ 0 & x \notin \mathbb{Q}. \end{cases} \] Let \(\left. f\right\rvert_{\mathbb{Q} } : \mathbb{Q} \to \mathbb{R}\) be the same function, restricted to the domain \(\mathbb{Q} \). Show that this restriction \(\left. f\right\rvert_{\mathbb{Q} }\) is continuous.

Problem 4.

Let \(f: \mathbb{R}\to \mathbb{R}\) be a differentiable function. Suppose that its derivative is bounded, i.e. that there exists \(M\in \mathbb{R}\) suc that \(\left\lvert f’(x) \right\rvert \leq M\) for all \(x\in \mathbb{R}\). Show that \(f\) is uniformly continuous.

Hint: Use the mean value theorem.

Problem 5.

Give an example of a differentiable function \(f: \mathbb{R}\to \mathbb{R}\) such that \(f\) is uniformly continuous but \(f’\) is not bounded. Explain why this does not contradict the previous problem.

Problem 6.

Let \(f:\mathbb{R}\to \mathbb{R}\). Recall that \(\lim _{x\to x_0} f\left( x \right) = L\) if for all \(\epsilon > 0\), there exists \(\delta > 0\) such that for all \(x\in \mathbb{R}\), \[0 < \left\lvert x - x_0 \right\rvert < \delta \implies \left\lvert f(x) - L \right\rvert < \epsilon .\]

1. Show that \(\lim _{x\to x_0}f(x) = L\) if and only if: for all sequences \(\left\lbrace x_n \right\rbrace\subseteq \mathbb{R}\setminus \left\lbrace x_0 \right\rbrace\) satisfying \(x_n \to x_0\), we have \(f\left( x_n \right) \to L\).
2. Show that \(f\) is continuous at \(x_0\) if and only if \(\lim _{x\to x_0}f(x) = f\left( x_0 \right)\).