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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 ??

Note by Ashwin Upadhyay
2 years, 1 month 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. Kuldeep Guha Mazumder · 2 years ago

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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 Vihari Vemuri · 2 years, 1 month ago

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@Vihari Vemuri I couldn't understand your first reason. Ashwin Upadhyay · 2 years, 1 month ago

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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. Azhaghu Roopesh M · 2 years, 1 month ago

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@Azhaghu Roopesh M Very good. He is fooling. Harendran Bala · 2 years, 1 month ago

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@Azhaghu Roopesh M 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. Ashwin Upadhyay · 2 years, 1 month ago

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@Ashwin Upadhyay 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 . Azhaghu Roopesh M · 2 years, 1 month ago

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@Azhaghu Roopesh M In fact i found a whole set of such apparent fallacies Ashwin Upadhyay · 2 years, 1 month ago

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@Ashwin Upadhyay Wow . You've been busy :P Azhaghu Roopesh M · 2 years, 1 month ago

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@Azhaghu Roopesh M BTW i just found another note on this Ashwin Upadhyay · 2 years, 1 month ago

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