>>10163389

>>10163768

You know what, there is something I quite like about this way of doing things, it motivates the definition of continuity. Let me explain. In order for [math]\frac{d(x^2)}{dx}[/math] to converge to something reasonable as [math]{dx}[/math] goes to zero, we need to put some conditions on [math]d(x^2)[/math] dependent on [math]{dx}[/math]. This is necessary since we want to extract some finite number from [math]\frac{d(x^2)}{dx}[/math] so we need to understand what [math]d(x^2)[/math] does as [math]{dx}[/math] goes to zero. Well, obviously we want [math]d(x^2)[/math] to go to zero, so using our definition of limits that means that for every [math]0 < \epsilon[/math] we have that [math]|d(x^2)|< \epsilon[/math] in the limit as [math]dx[/math] goes to zero. But hang on, we've just used another limit, so if we use the definition of what a limit means for [math]dx[/math] to go to zero it means that [math]|dx|< \delta[/math] where [math]\delta[/math] is some positive real. So tying this all together, in order for [math]\frac{d(x^2)}{dx}[/math] to make sense we need [math]d(x^2)[/math] to go to zero as [math]dx[/math] goes to zero. This means that [math]|d(x^2)|< \epsilon[/math] subject to the constraint that [math]|dx|< \delta[/math] for some [math]\delta[/math]. We may call this necessary condition continuity. But it is not sufficient, take for example the function [math]|x|[/math] at zero, this is not differentiable. I think this perspective might be useful. People are familiar with polynomials, they're easy, but a generic continuous function may be pathological as fuck. I think a neat way of motivating continuity may be to start with >>10163389 and then show that in order for this process to work appropriately one must introduce a certain condition that is equivalent to continuity, thus motivating the definition like I've hopefully done above, and then showing it is necessary but not sufficient for a function to be differentiable at a point.