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I would say continuity is the idea that points numerically close to the input point of a function agree with the value of the function at that point. Now, suppose I introduce the idea of a derivative, it takes two close by points of the function and computes the slope.

$$ \lim_{x \to a} f(x) = f(a)$$

However, if a function is continuous i.e: really close by points numerically agree, how can there be a variation in the function to define the derivative?

Excuse me if the question is poorly phrased.

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    $\begingroup$ I find that an important feature in demonstrating what something is is to also demonstrate what it is not. Particularly, continuity has a definition but it conceptually straightforward to demonstrate discontinuity and then conceptually define continuity as not discontinuous. $\endgroup$ – Carser Jun 6 at 18:12
  • $\begingroup$ How can a limit exist if a function is not constant? $\endgroup$ – Opal E Jun 7 at 18:11
  • $\begingroup$ This seems more like a math question than a question about teaching math. $\endgroup$ – Ben Crowell Jun 7 at 23:14
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Continuity

Method 1

One of the most intuitive ways that I like to think about what it means for a function to be "continuous" is by the "drawing test": can you draw the function's graph on a piece of paper without ever removing your pen/pencil from the paper? A continuous function is a function whose graph is an unbroken curve.

Method 2

Another way that might be helpful in understanding what it means for a function to be continuous is to actually look at the types of discontinuities for a function. You can read Wikipedia's article on the types of discontinuities for more information. In particular, there are three types of discontinuities:

  1. A removable discontinuity:
  1. A jump discontinuity:
  1. An infinite (or essential) discontinuity:

As you can see, in neither of these scenarios can we draw the function's graph without removing our pencil from the page. Hence, the function is not continuous.


Here's a nice visual summary of each discontinuity:

Differentiability

Differentiability is a more technical and lengthier topic than continuity, as differentiability brings about discussions regarding smoothness and multi-variability. I'll leave it up to you to teach/introduce differentiability to your students as per the specified curriculum and course outline. The topic is rich, as even in a typical Calculus 3 course, differentiability is studied in depth.

That being said, sticking to functions with a single variable, and proceeding in a fashion as we did above, we know intuitively now that a continuous function is one that has no "breaks" in its graph. Similarly, a differentiable function from $\mathbb{R}^2$ to $\mathbb{R}$ is a function that not only has no breaks in its graph, but also has a well-defined plane that is tangent to the graph at each point. Hence, we expect a differentiable function to not have any sharp folds, corners, or "peaks" in the graph; the graph must be smooth.

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  • $\begingroup$ While not disagreeing with your general approach, I would be wary of introducing the concept of an essential discontinuity at high-school, and would only do so for well-prepared students and with carefully chosen examples. Cases like f(x) = 1/x (which is implied by your picture) are best avoided, since f(x) = 1/x is in fact continuous everywhere it is defined. The canonical example of a function with an essential discontinuity is the piecewise defined function g(x) = sin(1/x) when x ≠ 0 and g(x) = x when x = 0. $\endgroup$ – Stephan Kubicki Jun 9 at 7:55
  • $\begingroup$ @StephanKubicki Furthermore, a true infinite discontinuity (an infinite discontinuity at a point in the domain) can often be regarded as a jump discontinuity, but I'm not sure that essential discontinuities are actually infinite discontinuities. $\endgroup$ – Ryan G Jun 12 at 16:37
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For continuity, I would generally follow one or both methods suggested by CuriosityCalls' answer (modulo my reservations about essential discontinuities, which I expressed in my comment to that answer).

For differentiability, in one sense this is straightforward for students who have already encountered the definition of the derivative as $\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$. In this case, $f(x)$ is differentiable just in case that limit exists.

Of course in practice, at the high-school level, matters are not so simple, since we often want to avoid having to compute that limit to determine whether the derivative exists. At this level, it is typically sufficient for students to know that derivatives have the intermediate value property, and hence a derivative can not have a jump discontinuity. This is enough to justify simple cases like $f(x)= |x|$ not having a derivative at $x=0$, and the intuitive idea that a function is not differentiable at a point if its graph has a "sharp corner".

However, to avoid incorrect generalisation, I would also consider showing students the function $g(x) = \left\{ \begin{array}{ll} x\sin\left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{array} \right. $

and observe that while the derivative does not have a jump discontinuity at 0, the derivative does not exist at 0. This is just to emphasise that what matters for whether a function is differentiable at $x=a$ is whether the limit of the difference quotient exists; showing that a (hypothetical) derivative is not continuous does NOT prove that the derivative does not exist. The test described above is just a handy shortcut method that can be used when the functions are suitably nice.

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