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\noindent{\LARGE Math 131BH, MS$^2$, Winter 01}
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\noindent{\Large Workshop \#1}
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1. Let $f:\mathbb{Q} \rightarrow \mathbb{R}$ be defined by $f(x)=0$ if 
$x<\sqrt{2}$ and $f(x)=1$ if $x>\sqrt{2}$. Show that $f$ is continuous on 
all of $\mathbb{Q}$.
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2. Define functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ by 
$$f_n(x)=\frac{nx}{1+|nx|}.$$ Show that $f_n$ is continuous for every 
$n\in \mathbb{N}$. For which $x\in \mathbb{R}$ is the function 
$$f(x)=\lim_{n\rightarrow \infty}f_n(x)$$ defined and continuous?
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3. Show that $f:\mathbb{R}_+ \rightarrow \mathbb{R}$ defined by 
$f(x)=\sqrt{x}$ is uniformly continuous, but the function 
$g:\mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x)=x^2$ is not.
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4. For any metric space $X$, is the identity function on $X$ 
continuos? Uniformly continuous? Prove your answers.
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5. Let $[a,b]$ be an interval in $\mathbb{R}$ and $f$ a continuous, 
real valued, one to one function on $[a,b]$. Prove that 
$f([a,b])=[f(a),f(b)]$ or $[f(b),f(a)]$ (whichever one makes sense).
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6. Let $f_1,f_2,...$ be real valued continuous functions on a metric 
space $X$ and assume for each $x\in X$, the sequence $(f_1(x),f_2(x),...)$ 
is bounded. Define $f(x)=sup(f_1(x),f_2(x),...)$ for each $x\in X$. Prove 
or find a counterexample: $f$ is continuous.


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