Is the statement valid? 
Now consider a point just next to point (,), say (+ℎ,′). Here, ℎ is infinitesimally small, such that ℎ→0. Now, the slope of the line between these 2 points will give us the instantaneous rate of change as after that instant we will move on the next instant/point.


Above statement is very important to understand. Point  and  are actually next to each other! (that's how we have defined). Even if you see a curve of 2 on your screen and take 2 pixels that are next to each other to be lying on the curve, they are still actually not next to each other with respect to the curve as between those 2 pixels there are infinitely many points but the points that we have taken, that is,  and  are next to each other!

The above two paragraphs are from an answer on this forum link to answer. I see flaws in the answer, but I was confused since it was getting upvotes. My question is, how can you take two points next to each other? Doesn't that violate the density property of real numbers?
For context, the answer was written to explain what derivatives are and what they mean. I just feel the statements used are mathematically invalid; if they are then the answer should be removed as it would mislead many. Let me know your thoughts.
 A: I've spent a while trying to find the correct wording for this, as I don't want to criticize the answerer for doing their best to help someone else -- teaching mathematics is a whole other challenge! I don't really like the presentation of $P$ and $H$ as "next to each other," but for a different reason. Certainly $x+h$ and $x$ cannot be 'next to each other' when $h$ is nonzero for the reason you have stated, but in a secondary school setting I don't think the density of $\mathbb{R}$ is much helpful pedagogically speaking.
I think a good way to approach this topic is in keeping with a philosophy of "start where your learner is familiar and take steps into the unfamiliar from there." If you take a simple quadratic and pick a point on it, students familiar with the concept of a tangent line will have a pretty good idea of what it should look like. Then you ask the students to tell you about the tangent line and its properties. What is the slope of the line? What is its equation? Your students may feel dumbfounded: surely this line has to have an equation and a slope... what are they? Then after a good discussion you can start with the construction of tangent lines using secant lines between points whose distance is progressively smaller, introducing the concepts of differential calculus.
At least to me, saying that $x$ and $h$ are "next to each other" seems to me to skip the important details of the above approach: instead of making it clear where the $x$ and $x+h$ came from, you're essentially saying, "I have two points: $(x,y_1)$ and $(x+h,y_2)$, and they're, like, really close to each other." This is problematic because it (a) runs into arguments about what "close" means and (b) might give students the false belief that $h$ is a specific number rather than a quantity that approaches zero. This could be remedied by saying something like

Consider two points $P(x,f(x))$ and $Q(x+h,f(x+h))$. Say $h$ is a positive, real number. When $h$ is quite large, the secant line $PQ$ won't be very representative of the tangent line at $P$. If you want $PQ$ to look more like the tangent line at $P$, you make $h$ smaller and smaller. Of course, just going straight to $h=0$ isn't helpful because we just get the same two points -- instead, you make $h$ smaller and realize that it only seems natural that the tangent line occurs when you make a secant line $PQ$ with $h=0$.

With regard to removing the answer, I don't think that would be justified at all. Not only does the answer not violate any of the rules, but it can also help spawn discussion of the sort that we are having right now, which I think is very useful to both learners and teachers alike. If you don't think the answer does a good job, you're free to downvote it or leave a comment suggesting improvement.
