Continuous function

**iLikeMaths** · March 5th 2009, 12:58 AM

prove using the $\epsilon , \delta$ definition of continuity that $f(x) = x^3 - 2x^2 + 1$ is continuous at $x=3$

this is the start of my solution, however i need help complting it

$f(x) = x^3 -2x^2 +1$
$c=3$
$f(c)= 27-18+1 = 10$

for each positive number $\epsilon$
$|f(x) -f(c)| < \epsilon$ for all $x$ satisfying $|x-c|<\delta$

and then i get

$|x-3||x^2 + x +3|<\epsilon$ for all $|x-3|<\delta$

so my question is what is the value of $\delta$ , can i just use any random number or does it have to be specific?

**Plato** · March 5th 2009, 04:21 AM

Originally Posted by iLikeMaths

prove using the $\epsilon , \delta$ definition of continuity that $f(x) = x^3 - 2x^2 + 1$ is continuous at $x=3$

this is the start of my solution, however i need help complting it

$f(x) = x^3 -2x^2 +1$
$c=3$
$f(c)= 27-18+1 = 10$

for each positive number $\epsilon$
$|f(x) -f(c)| < \epsilon$ for all $x$ satisfying $|x-c|<\delta$

and then i get

$|x-3||x^2 + x +3|<\epsilon$ for all $|x-3|<\delta$

so my question is what is the value of $\delta$ , can i just use any random number or does it have to be specific?

Assumming your algebra is correct (I did not check it)
$\begin{gathered} \left| {x - 3} \right| < 1\; \Rightarrow \;2 < x < 4 \hfill \\ \left| {x^2 + x + 3} \right| \leqslant \left| {x^2 } \right| + \left| x \right| + \left| 3 \right| < 16 + 4 + 3 = 23 \hfill \\ \delta = \min \left\{ {1,\frac{\varepsilon } {{23}}} \right\} \hfill \\ \end{gathered}$

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