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I just want to get a better grasp of this concept. I don't think you can, for example $F(x) = 1000x$.

If I want to be within $1000$ of $f(x)$, i.e. $\epsilon = 1000$, then $\delta$ would be $250$.

So, $f(249) = 249,000$ - which is not within $1,000$ of $x$, if $x = 1$.

Is this correct?

Thanks!

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    It is a well-known canard that Freshmen believe all functions to be linear. For example, $1/(x+y) = 1/x + 1/y$ is obvious, no?2012-06-01

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If $f$ is linear, $f(x)=mx+b$ and you can take $\delta=\frac{\epsilon}{ |m|}$. You should plug this into the $\delta - \epsilon$ definition to see that it is true. For differentiable $f$, you can usually use something close to $\frac \epsilon {f'}$ (note that in the previous case $f'=m$) but you might need to use something smaller. This comes from the fact that the derivative gives the best local linear approximation, but higher order terms may be a problem.