The Slope

From the same source we have used on the last post comes:

If a function is a set of ordered pairs, it doesn't change, so that there is no rate of change is insane: Any Cartesian function will be a set of ordered pairs, and plenty of them change, like we don't really know what the critic meant by change, but we assume it is change in height. Even the constant function has a rate of change, regardless: That is zero. For instance, f(x)=5 gives you f'(x)=0. It does have a rate of change. It is just that the rate is zero. 

Perhaps what they both meant, critic and writer, is that the slope of the function at a given point is given by the derivative of the function on that point, and the derivative function has the shape of a line that is tangent to the point. Therefore, the rate of change of the function, which is the slope of the function, on a point is given by the derivative on that point. The slope of the own derivative would be something that shouldn't be relevant here. 

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One Place Function?

Still from the same document we saw on the other post comes this one:

So, as said before, i should be a reserved letter in Mathematics, so that it shouldn't mean increment, not really. Maybe that is what they have done here, but they shouldn't have done that, that is all. You don't take the symbol for the set of the real numbers, which is the R with double strike, and insert it somewhere to mean variable, you simply don't. The name gives you no clue as to the nature of the thing named, he says, that making the criticism. Yet, in Mathematics, we should try to give clues also through the names we give to things. 

One-place function is not a good name. Perhaps they meant a real function. If so, they should just have said that: A real function or an R^2 function or a Cartesian function. A two-place function could then be an R^3 function (you have two coordinates in the domain, so that you use those to get your result, and that would then form the third coordinate), we suppose. 

The occurrence of the variable is free is a dodge claim, given the context. We imagine he refers to Logic and no quantifiers are noticed. We never saw a variable that is the name of a number. We suppose he meant the variable, in this case, is not a place holder for a number. If not, then we don't know what a variable is anymore, we suppose. Now, what function? If there is no equal sign, we don't really have a function there. Again, x and x+i are not points: They are at most coordinates or elements of the domain. 

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Points and Coordinates

Someone sent me the extract below to criticise:

Points? x, x+i could at most be coordinates of points instead, is it not? A point in the Cartesian plane would have two coordinates: one to tell us until where we walk in the horizontal direction and another to tell us until where we walk in the vertical direction to reach the point. 

i should be regarded as a reserved letter, since, with it, we denote imaginary, the square root of minus one. In this case, we only use it if we really mean that. They couldn't have meant imaginary there because they are talking about two generic points instead.

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