Let us see an erroneous deduction: $$x^2=x^2,~\forall x \in \Bbb R,$$ $$\Leftrightarrow x^2=x \cdot x,~\forall x \in \Bbb R ,$$ $$\Rightarrow x^2=\underbrace{x+x+...+x}_{x ~\text{times}},~\forall x \in \Bbb N ,$$ I want to know the logical deception in differentiating the equation to get, $$2x=x,~\forall x \in \Bbb N ,$$ $$\Rightarrow 2=1.$$
Is it a problem of discrete differentiability?, as I feel the forward difference of $x^2$ is $D_{+}(x^2)=\frac{(x+1)^2-x^2}{1}=2x+1$ which is not equal to the backward difference of $D_{-}(x^2)=\frac{x^2-(x-1)^2}{1}=2x-1$.
OR
Can I claim that differentiation cannot be sensible in a non-manifold structure $\Bbb N$?
