# Is it a fallacy ??

Recently, I was doodling in my rough notebook and found something strange:

Let x belong to N (natural no.)

$$x^{2}$$ = $$x^{2}$$

$\Rightarrow$ $x^{2}$ = $x+x+x+x+x+x....(x times)$

(Differentiating on both sides)

$2x$ = $1+1+1+1+1+1+1....(x times)$

$\Rightarrow 2x = x$

Where was I wrong ??

5 years, 10 months ago

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When you are defining x^2=x+x+..+x(x times), you are tacitly defining the domain of f(x) to be N, the set of natural numbers. However, f(x) cannot be differentiable in the domain of N. This is because, if a function f defined from a domain D (a subset of R, the set of reals) to R (the set of reals) has to be differentiable at a real point c, then a necessary criterion is that c has to be a limit point of D and c has to be an element of D itself. In other words, c has to be a member of D such that every arbitrarily small neighbourhood of c has an element of D other than c. But no natural number is a cluster point of the set N. Here lies the fallacy.

- 5 years, 8 months ago

Actually you cannot differentiate the function f(x) = $x^{2}$ when you have selected the Domain as Natural numbers . After all , the Natural numbers as a domain will comprise just discrete points , so it'll not be differentiable.

- 5 years, 10 months ago

Firstly I differentiated it generally but in that case i couldn't write the no. x times as x could be fraction or 0. Thank you now i have understood.

- 5 years, 10 months ago

Anytime . Btw there was a question on this fallacy posted by @Sandeep Bhardwaj sir , I'm not able to find it . I'll give you the link if I'm able to find it .

- 5 years, 10 months ago

BTW i just found another note on this

- 5 years, 10 months ago

In fact i found a whole set of such apparent fallacies

- 5 years, 10 months ago

Wow . You've been busy :P

- 5 years, 10 months ago

Very good. He is fooling.

- 5 years, 10 months ago

When you are differentiating on both sides, you assumed that in x times the x is constant. Also you can't talk abt differention, of non continuous function

- 5 years, 10 months ago

I couldn't understand your first reason.

- 5 years, 10 months ago