Exercise 4.3.2

Question

To gain a deeper understanding of the relationship between $\epsilon$ and $\delta$ in the definition of continuity, let's explore some modest variations of Definition 4.3.1. In all of these, let $f$ be a function defined on all of $\mathbf{R}$. \begin{enumerate}[label=(\alph*)] \item Let's say $f$ is \emph{onetinuous} at $c$ if for all $\epsilon>0$ we can choose $\delta=1$ and it follows that $|f(x)-f(c)|<\epsilon$ whenever $|x-c|<\delta$. Find an example of a function that is onetinuous on all of $\mathbf{R}$. \item Let's say $f$ is \emph{equaltinuous} at $c$ if for all $\epsilon>0$ we can choose $\delta=\epsilon$ and it follows that $|f(x)-f(c)|<\epsilon$ whenever $|x-c|<\delta .$ Find an example of a function that is equaltinuous on $\mathbf{R}$ that is nowhere onetinuous, or explain why there is no such function. \item Let's say $f$ is \emph{lesstinuous} at $c$ if for all $\epsilon>0$ we can choose $0<\delta<\epsilon$ and it follows that $|f(x)-f(c)|<\epsilon$ whenever $|x-c|<\delta$. Find an example of a function that is lesstinuous on $\mathbf{R}$ that is nowhere equaltinuous, or explain why there is no such function. \item Is every lesstinuous function continuous? Is every continuous function lesstinuous? Explain. \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item The constant function $f(x) = k$ is onetinuous, in fact it is the only onetinuous function (Think about why)
  \item The line $f(x) = x$ is equaltinuous
  \item $f(x) = 2x$ is lesstinuous but nowhere-equaltinuous
  \item Every lesstinuous function is continuous, since the definition of lesstinuous is just continuous plus the requirement that $0 < \delta < \epsilon$.

And every continuous function is lesstinuous since if $\delta > 0$ works we can set $\delta' < \delta$ and $\delta' < \epsilon$ so that $|x-c| < \delta' < \delta$ still implies $|f(x)-f(c)| < \epsilon$
   \end{enumerate}

Exercise 4.3.2

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Exercise 4.3.2

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