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 ??

#MathematicalFallacies

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

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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 - 6 years 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.

A Former Brilliant Member - 6 years, 1 month 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.

Ashwin Upadhyay - 6 years, 1 month ago

Very good. He is fooling.

Harendran Bala - 6 years, 1 month 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

Vihari Vemuri - 6 years, 1 month ago

I couldn't understand your first reason.

Ashwin Upadhyay - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$