VIII. Continuity #

Problem 1 #

Suppose that \((S, \rho)\) is a metric space and let \(f: S \to S\) be the identity function defined by \(f(s)=s\) for all \(s \in S\). Show that \(f\) is continuous by using our original definition of continuity.

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Problem 2 #

Suppose that \((S, \rho)\) is a metric space and let \(f: S \to S\) be a contraction mapping, so that there is a positive constant \(\alpha < 1\) such that for all distinct \(x,y \in S\), we have

$$\rho(f(x), f(y)) \leq \alpha \, \rho(x,y).$$

Show that \(f\) is a continuous function.

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Problem 3 #

Show that the absolute value function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = |x|\) is continuous.

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Problem 4 #

The original definition of a continuous function is fairly cumbersome for daily use. The goal of this exercise is to verify a somewhat more palatable characterization of continuous functions. Let \((S_1,\rho_1)\) and \((S_2,\rho_2)\) be metric spaces and consider \(f:S_1 \to S_2\). Show that \(f\) is continuous iff for every open set \(U \subset S_2\), the inverse image \(f^{-1}(U)\) in \(S_1\) is also an open set.

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Problem 5 #

Consider a function \(f: S \to \mathbb{R}\) from some metric space \((S,\rho)\) into the real numbers. Define another function \(|f|: S \to \mathbb{R}\) by letting

$$|f|(x) = |f(x)| \qquad \forall x \in S.$$

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Problem 6 #

Consider a function \(f:S_1 \to S_2\). Show that if \(x\) is not a limit point of \(S_1\), then \(f\) must be continuous at \(x\).

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Problem 7 #

Show that every polynomial \(p: \mathbb{R} \to \mathbb{R}\) is a continuous function.

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Problem 8 #

Consider the function \(f: \mathbb{R}^2 \to \mathbb{R}\) defined by

$$f(x,y) = \begin{cases} \dfrac{xy}{x^2+y^2} & \text{if } (x,y) \neq (0,0), \\[6pt] 0 & \text{if } (x,y) = (0,0). \end{cases}$$

Determine whether \(f(x,y)\) is continuous at the origin \((0,0)\) and justify your answer.

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Problem 9 #

Consider \(\lim_{x \to 0} x\sin\left(\tfrac{1}{x}\right)\). It seems very reasonable that this limit should equal zero: \(x\) itself approaches zero and \(\sin\left(\tfrac{1}{x}\right)\) is always between negative one and one, so their product should also approach zero. While this is great intuition, it is not much of a proof. Use the formal \(\epsilon\)-\(\delta\) definition to show that indeed

$$\lim_{x \to 0} x\sin\left(\tfrac{1}{x}\right) = 0.$$

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Problem 10 #

Consider the function \(f: \mathbb{R} \to \mathbb{R}\) defined by

$$f(x) = \begin{cases} \dfrac{1}{q} & \text{if } x=\dfrac{p}{q} \in \mathbb{Q} \text{ in lowest terms, and} \\[6pt] 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \text{ and } x=0. \end{cases}$$

Prove that \(f\) is continuous at \(x=0\) as well as at every irrational number.

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