CRN 11378: HW 4

Problem 25. Let $f:[a,b]\to\mathbb{R}$ be an increasing function. Show that $\lim_{x\to a}f(x)$ exists. What can you say about the relationship between this limit and $f(a)$?

Problem 26. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that $h$ is continuous on $\mathbb{R}$.

Problem 27. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$, and assume that for all $\varepsilon>0$ there is an $N>0$ such that $|f(x)|<\varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $\mathbb{R}$.

Problem 28. Let $f:[a,b]\to\mathbb{R}$ be a function. We say $f$ satisfies $(*)$ if there is an $M>0$ such that $|f(x)-f(y)|\leq M\cdot |x-y|$ for all $x,y\in [a,b]$.

1. Let $g:[0,1]\to\mathbb{R}$ be given by $g(x)=\sqrt{x}$. Show that $g$ does not satisfy $(*)$.
2. Is $g$ uniformly continuous on $[a,b]$? Is $g$ uniformly continuous on $(a,b)$? Explain!