91Ó°ÊÓ

Let \(S \subseteq \mathbb{R}\) and suppose there exists a sequence \(\left(x_{n}\right)\) in \(S\) that converges to a number \(x_{0} \notin S .\) Show that there exists an unbounded continuous function on \(S\).

Let \(E\) and \(F\) be connected sets in some metric space. (a) Prove that if \(E \cap F \neq \varnothing,\) then \(E \cup F\) is connected. (b) Give an example to show that \(E \cap F\) need not be connected. Incidentally, the empty set is connected.

Let \(f\) and \(g\) be real-valued functions. (a) Show that \(\min (f, g)=\frac{1}{2}(f+g)-\frac{1}{2}|f-g|.\) (b) Show that \(\min (f, g)=-\max (-f,-g).\) (c) Use (a) or (b) to prove that if \(f\) and \(g\) are continuous at \(x_{0}\) in \(\mathbb{R},\) then \(\min (f, g)\) is continuous at \(x_{0}.\)

Suppose that \(f\) is a real-valued continuous function on \(\mathbb{R}\) and that \(f(a) f(b)<0\) for some \(a, b \in \mathbb{R} .\) Prove that there exists \(x\) between \(a\) and \(b\) such that \(f(x)=0\)

Prove that a polynomial function \(f\) of odd degree has at least one real root. Hint: It may help to consider first the case of a cubic, i.e., \(f(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\) where \(a_{3} \neq 0\)

We say a function \(f\) maps a set \(E\) onto a set \(F\) provided \(f(E)=F\) (a) Show that there is a continuous function mapping the unit square $$ \left\\{\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}: 0 \leq x_{1} \leq 1,0 \leq x_{2} \leq 1\right\\} $$ onto [0,1] (b) Do you think there is a continuous function mapping [0,1] onto the unit square?

Suppose that \(f\) is continuous on [0,2] and that \(f(0)=f(2) .\) Prove that there exist \(x, y\) in [0,2] such that \(|y-x|=1\) and \(f(x)=f(y)\) Hint: Consider \(g(x)=f(x+1)-f(x)\) on [0,1]

Show that there exist continuous functions (a) mapping (0,1) onto [0,1] (b) mapping (0,1) onto \(\mathbb{R}\) (c) mapping [0,1]\(\cup[2,3]\) onto [0,1]

Find the following limits. (a) \(\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{x-a}\) (b) \(\lim _{x \rightarrow b} \frac{\sqrt{x}-\sqrt{b}}{x-b}, b>0\) (c) \(\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x-a}\)