Chapter 3

a) If \(f:[0,1] \rightarrow \mathbb{R}\) is given by \(f(x):=x^{m}\) for an integer \(m \geq 0,\) show \(f\) is Lipschitz and find the best (the smallest) Lipschitz constant \(K\) (depending on \(m\) of course). Hint: \((x-y)\left(x^{m-1}+x^{m-2} y+x^{m-3} y^{2}+\cdots+\right.\) \(\left.x y^{m-2}+y^{m-1}\right)=x^{m}-y^{m}\) b) Using the previous exercise, show that if \(f:[0,1] \rightarrow \mathbb{R}\) is a polynomial, that is, \(f(x):=a_{m} x^{m}+a_{m-1} x^{m-1}+\) \(\cdots+a_{0},\) then \(f\) is Lipschitz.

True/False, prove or find a counterexample. If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function such that \(\left.f\right|_{\mathbb{Z}}\) is bounded, then \(f\) is bounded.

Suppose \(S \subset \mathbb{R}\) and \(c\) is a cluster point of S. Suppose \(f: S \rightarrow \mathbb{R}\) is bounded. Show that there exists a sequence \(\left\\{x_{n}\right\\}\) with \(x_{n} \in S \backslash\\{c\\}\) and \(\lim x_{n}=c\) such that \(\left\\{f\left(x_{n}\right)\right\\}\) converges.

Suppose \(f:[-1,0] \rightarrow \mathbb{R}\) and \(g:[0,1] \rightarrow \mathbb{R}\) are continuous and \(f(0)=g(0) .\) Define \(h:[-1,1] \rightarrow \mathbb{R}\) by \(h(x):=f(x)\) if \(x \leq 0\) and \(h(x):=g(x)\) if \(x>0 .\) Show that h is continuous.

Prove that the Dirichlet function \(f:[0,1] \rightarrow \mathbb{R}\) defined by \(f(x):=1\) if \(x\) is rational and \(f(x):=0\) otherwise cannot be written as a difference of two increasing functions. That is, there do not exist increasing \(\mathrm{g}\) and \(h\) such that, \(f(x)=g(x)-h(x) .\)

Suppose for \(f:[0,1] \rightarrow \mathbb{R}\) we have \(|f(x)-f(y)| \leq K|x-y|\) for all \(x, y\) in \([0,1],\) and \(f(0)=f(1)=0 .\) Prove that \(|f(x)| \leq K / 2\) for all \(x \in[0,1] .\) Further show by example that \(K / 2\) is the best possible, that is, there exists such a continuous function for which \(|f(x)|=K / 2\) for some \(x \in[0,1] .\)

Suppose \(f:[0,1] \rightarrow(0,1)\) is a bijection. Prove that \(f\) is not continuous.

Suppose \(g: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function such that \(g(0)=0,\) and suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is such that \(|f(x)-f(y)| \leq g(x-y)\) for all \(x\) and \(y .\) Show that \(f\) is continuous.

Suppose \(f:(a, b) \rightarrow(c, d)\) is a strictly increasing onto function. Prove that there exists a \(g:(a, b) \rightarrow(c, d),\) which is also strictly increasing and onto, and \(g(x)

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and periodic with period \(P>0\). That is, \(f(x+P)=f(x)\) for all \(x \in \mathbb{R}\). Show that \(f\) is uniformly continuous.

Show that the condition of being a cluster point is necessary to have a reasonable definition of a limit. That is, suppose \(c\) is not a cluster point of \(S \subset \mathbb{R},\) and \(f: S \rightarrow \mathbb{R}\) is a function. Show that every \(L\) would satisfy the definition of limit at \(c\) without the condition on \(c\) being a cluster point.

Challenging): Suppose \(f(x+y)=f(x)+f(y)\) for some \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that \(f\) is continuous at \(0 .\) Show that \(f(x)=\) ax for some \(a \in \mathbb{R} .\) Hint: Show that \(f(n x)=n f(x),\) then show \(f\) is continuous on \(\mathbb{R}\). Then show that \(f(x) / x=f(1)\) for all rational \(x\).

Suppose \(f: S \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow[0, \infty)\) are functions, \(g\) is continuous at \(0, g(0)=0,\) and whenever \(x\) and \(y\) are in \(S\) we have \(|f(x)-f(y)| \leq g(|x-y|) .\) Prove that \(f\) is uniformly continuous.

Suppose \(g(x)\) is a monic polynomial of even degree \(d,\) that is, $$ g(x)=x^{d}+b_{d-1} x^{d-1}+\cdots+b_{1} x+b_{0} $$ for some real numbers \(b_{0}, b_{1}, \ldots, b_{d-1} .\) Show that \(g\) achieves an absolute minimum on \(\mathbb{R}\).

Suppose \(S \subset \mathbb{R}\) and let \(f: S \rightarrow \mathbb{R}\) and \(g: S \rightarrow \mathbb{R}\) be continuous functions. Define \(p: S \rightarrow \mathbb{R}\) by \(p(x):=\max \\{f(x), g(x)\\}\) and \(q: S \rightarrow \mathbb{R}\) by \(q(x):=\min \\{f(x), g(x)\\} .\) Prove that \(p\) and \(q\) are continuous.

Suppose \(f(x)\) is a polynomial of degree \(d\) and \(f(\mathbb{R})=\mathbb{R} .\) Show that \(d\) is odd.

Suppose \(f:[-1,1] \rightarrow \mathbb{R}\) is a function continuous at all \(x \in[-1,1] \backslash\\{0\\} .\) Show that for every \(\varepsilon\) such that \(0<\varepsilon<1,\) there exists a function \(g:[-1,1] \rightarrow \mathbb{R}\) continuous on all of [-1,1] , such that \(f(x)=g(x)\) for all \(x \in[-1,-\varepsilon] \cup[\varepsilon, 1]\), and \(|g(x)| \leq|f(x)|\) for all \(x \in[-1,1]\).

(Challenging): A function \(f: I \rightarrow \mathbb{R}\) is convex if whenever \(a \leq x \leq b\) for \(a, x, b\) in \(I,\) we have \(f(x) \leq f(a) \frac{b-x}{b-a}+f(b) \frac{x-a}{b-a} .\) In other words, if the line drawn between \((a, f(a))\) and \((b, f(b))\) is above the graph of \(f\). a) Prove that if \(I=(\alpha, \beta)\) an open interval and \(f: I \rightarrow \mathbb{R}\) is convex, then \(f\) is continuous. b) Find an example of a convex \(f:[0,1] \rightarrow \mathbb{R}\) which is not continuous.