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\head {\bf MATH 165 FINAL\rm} \endhead
\head {\bf 12/16/02}\endhead
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\noindent NAME: 
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\noindent 1) (30 pts.) Write a proof of the following theorem (state the theorem here):
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\noindent 2) (20 points) Give the $\epsilon$- $\delta$ definition of continuity of $f:\Bbb R\to \Bbb R$ at a point $c\in \Bbb R$.  
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\noindent 3) (20 pts.) Define what is meant by denseness of the rationals (as well as irrationals) on the real line. Is denseness a consequence of the
purely algebraic axioms for $\Bbb R$? Explain.
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\noindent 4) (20 pts.) Let $f:\Bbb R \to \Bbb R$ be given by $f(x)=0$ if $x\leq 0$ and $f(x) =\sqrt x$ if $x>0$. Using the $\epsilon$- $\delta$ definition
of continuity, show that $f$ is continuous at $a=0$.
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\noindent 5) (20 pts.) State the principle of mathematical induction and prove that for every $n\in \Bbb N$ one has
$$ 1+2+3+... +n= {n(n+1) \over 2}.$$
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\noindent 6) (20 pts) a) Using the formula $\sin (a+b) -\sin (a-b)=2\sin a \cos b$
show that 
$$ \int_0^{2\pi} \sin nx \;\cos mx \;\; dx =0,$$
where $m, \;n \in \Bbb Z$.
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\noindent b) Compute 
$$ \int_0^{2\pi} (\sin 3x + \cos 3x)^2 dx.$$
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\noindent 7) (20 pts.) Recall that every rational number $x$ can be written uniquely as $x= {m \over n}$, where $m$ and $n$ are integers, $n>0$, and the
fraction is in its lowest terms. Define  $f:\Bbb R \to \Bbb R$ by $f(x)=0$ if $x$ is irrational and $f(x)={1\over n}$ if $x= {m \over n}$ is a rational
number written as above.
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\noindent a) Show that $f$ is continuous at every irrational number. \it Hint \rm : Show first that for any $M>0$, any bounded interval contains only
finitely many rationals with denominator $n\leq M$. Next, think about what it means to say that "$\delta$ cannot be found" and use denseness of the
rationals...
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\noindent b) Show that $f$ is discontinuous at every rational number. \it Hint \rm : Use denseness of the irrationals.
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