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\noindent {\bf Question}

\noindent Define what it means for a function $f: {\bf
R}\rightarrow {\bf R}$ to be continuous.  Using the definition,
show that the function $f(x) = 2x-5$ is continuous.

\medskip

\noindent {\bf Answer}

\noindent $f$ is continuous at $a$ if $\lim_{x\rightarrow a} f(x)
=f(a)$. $f$ is continuous if it is continuous at every point in
its domain.

\medskip
\noindent To show that $f(x) = 2x-5$ is continuous, we show that
it is continuous at $a$ for every $a$.  That is, we need to show
that
\[ \lim_{x\rightarrow a} (2x-5) =2a-5. \]
So, for any $\varepsilon >0$, take $\delta
=\frac{1}{2}\varepsilon$. Then, if $| x-a| < \delta =\frac{1}{2}
\varepsilon$, then
\[ |f(x) -f(a) | =| (2x-5) -(2a-5)| = 2|x-a| < 2\frac{1}{2}\varepsilon
=\varepsilon, \] and so the definition of $\lim_{x\rightarrow a}
f(x) =f(a)$ is satisfied.



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