> So, as you restrict the values to integers, you are basically making the func discontinuous.
No, I restrict N to integers but not x, and this is where you keep going wrong.
Saswat, is f(x)=x discontinuous? What about f(x)=2x? What about f(x)=3x? You see, this is a FAMILY of STRAIGHT LINES, generalised as f(x)=Nx.
Let's take one member of this family as an example: f(x)=3x. So N=3. We draw f(x) out as a straight line. You insist you know what that looks like (despite claiming it to be discontinuous) so I won't bother with a screenshot. Now, look carefully. The line passes through a number of points where x has a value and y has a value, and the line relates x-values to y-values. Where x=0, y=0, for example, and where x=1, y=3; where x=10, y=30; where x=123, y=369. And so on, and as it's continuous, where x=pi, y=3pi; where x=sqrt(2), y=3sqrt(2) and there's a whole host of real numbers we could add to the list.
Now, look even more carefully at this next bit. Where x=3, y=9. Now, don't go off on this "haha, that proves f(x)=x^2" stuff again. We're still on the straight line f(x)=3x. x=3, y=9, because y=3*x, which is 3*3, which is 9 (which, barring major accidents, you should be able to figure out with your fingers). Now, because N=3, (take this slowly because we are STILL on the straight line y=3x), when x=3, (startling conclusion following) x equals N! (We're still on y=3x here, remember, which is a continuous straight line function.) Now, just because x=N at that point does not mean the whole line is now defined as y=x^2.
Now, the same is true for all other members of the same family. For the line y=2x, N=2, and where x=2, x=N, although the value of N has changed. Note that when x is some value other than 2, including all positive integers, x=N is not a true statement. N is a constant for the straight line, so x=N only where x=2. Where x=3, x no longer =N because 3!=2. And where x=4 (and N still hasn't changed, because if you can remember stuff longer than a goldfish then you'll know we're still talking about y=2x here) then x still doesn't =N, because x=4 and N=2 and 4!=2.
So let's recap what I did.
For y=x, N=1, and we're interested in the point along the straight line where x=1 (and because N=1, x=N, or N=x, which is actually the same thing).
For y=2x, N=2, and we're interested in the point along the straight line (which is a different straight line than in the previous statement) where x=2, and again because N=2 in this line, x=N and N=x.
For y=3x, N=3 and ... x=3 so x=N.
For y=4x, N=4 and ... x=4 so x=N
For y=5x, N=5 and ... x=5 so x=N
For y=6x, N=6 and ... x=6 so x=N
Got it yet? I'll add a couple more just in case you still haven't spotted the pattern yet.
For y=7x, N=7 and ... x=7 so x=N
For y=8x, N=8 and ... x=8 so x=N
These are all straight line graphs. Not one of them is a curve. If you try every positive integer I bet you won't find a single member of this family that is a curve, or that is discontinuous.
> Further ... f(x) = x + x + ... + (x times) is NOT a straight line, actually it's same as x^2.
Well, if you look back you'll see that because of your insistence it can only mean that, I've clarified my original intention as f(x)=x+x+x...(N times). You still don't appear to have figured that out. I'm happy to accept that you're completely unable to understand f(x)=x+x+x...(x times) means anything other than f(x)=x^2 despite 10 pages of clarification and insistence that that is not what I meant.
So this is why I've stopped saying f(x)=x+x+x...(x times) and have started saying, actually since quite early on, that f(x)=x+x+x...(N times). So if possible could you please stop talking about f(x)=x+x+x...(x times) because that is no longer the topic of discussion; we are now talking about f(x)=x+x+x...(N times).
In fact you did seem to have got the point on page 3:
Nope, that's where you go wrong.
f'(x) = x as long as N is a number independent of x.
N _is_ independent of x, that's what I've been saying all along. What gets you confused is the point along each straight line where the value of x corresponds to the value of N which is just a long way of saying x=N. So in the straight line graph y=4x there is a point where x=4 and y=16, and despite this, the graph is still y=4x and has not suddenly mutated into y=x^2. And if we generalise this there is a point along the straight line graph y=Nx where x=N and y=N*N.
But as soon as I generalise this that is where you completely lose the plot and go off on this "f(x)=x^2" stuff, insisting that N must be dependent on x, just because at some point in the graph x=N. That is your mistake.