In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition
$f$ is continuous when its graph can be drawn without lifting the pen.
I guess this is the intuition many students start with when they first encounter continuity. However, this isn't the formal definition of continuity.
Thus, my question is: How can we motivate the formal definition of continuity starting from the above intuition?
To clarify my question: Imagine yourself in a time when there was no formal definition of continuity (when for example people thought that functions fulfilling the intermediate value property are continuous). How would you argue that the formal definition of continuity must be of the form we have it today?
My attempt for answering the question: From the above intuition one may deduce several properties, continuous function should have:
- Continuous functions should fulfill the intermediate value property.
- The graph of a continuous function should be a connected subset of $\mathbb R^2$. (Although this is not true for continuous functions in general, this property makes sense when one starts from the above intuition)
- Small changes in the input result in small changes in the output.
Now we take this property as the definition of continuity which is (in our opinion) the most characteristic property of continuity. Here we might chose property (3) from which the $\epsilon$-$\delta$-definition can be deduced (note that (3) is similar to the definition of continuity by Cauchy which is one of the first formal definitions of continuity).
What do you think about this kind of explanation? When it is in your point of view a good explanation: Why should one use property (3) as the most characteristic property for continuity and not (1) or (2) or another property?