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Often, people are introduced to the notion of continuity by the idea that a function is continuous if and only if its graph can be drawn without lifting the pencil from the paper.

Now, after joining an undergraduate course and learning continuity formally using $\epsilon$-$\delta$ definition, I do not see any direct correlation between the formal definition and the intuitive idea about not having to lift the pencil to draw the graph.

How can we mathematically formalize the idea of "drawing the graph without lifting the pencil", and is it actually equivalent to the formal definition of continuity?

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    $\begingroup$ Consider the archetypal discontinuity generated by "lifting the pencil", i.e. a jump discontinuity. Think about how and why this fulfills the negation of continuity at that point. $\endgroup$
    – kodiak
    Feb 26 at 16:45
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    $\begingroup$ But what about other kinds of discontinuities other than jump discontinuity? $\endgroup$ Feb 26 at 16:48
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    $\begingroup$ Some version of this question has been asked many times before. Please have a look at these questions and answers and use the search bar too! $\endgroup$ Feb 26 at 17:00
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    $\begingroup$ Although this probably doesn't address your intent, it does answer the literal question: math.stackexchange.com/questions/1173342/… $\endgroup$ Feb 26 at 17:04
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    $\begingroup$ @AndrewD.Hwang Yes, I think that is a good duplicate target for this question. Good find. $\endgroup$
    – Xander Henderson
    Feb 26 at 17:05

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When first learning analysis, I always thought it was easier and slightly less abstract to consider the negation of a function being continuous (it being discontinuous) at a point.

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is discontinuous at $x_0 \in \mathbb{R}$ if there exists some $\varepsilon > 0$ such that for every $\delta > 0$ we choose, there exists at least one $x \in \mathbb{R}$ such that $$|x - x_0| < \delta \implies |f(x) - f(x_0)| \geq \varepsilon.$$

So, this gives us something to "work with" when considering the analogy of typical discontinuities (i.e., pencil test) which are taught in primary schooling. For instance, if we consider a jump discontinuity where we have to "pick up our pencil" to draw a continuous line, it is clear that there is some distance $\varepsilon > 0$ in the "$y$-direction" such that for any possible distance in the "$x$-direction," there will always be a "jump" between $f(x)$ and $f(x_0)$.

Same idea goes for other types of discontinuities.

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