I found this proof of $2 = 1$ online and as usual tried to debug it.
Consider the following true statement:
$x^2 = x + x + x + ... + x$ ($x$ times)
If we differentiate both sides, we get:
$$2x = 1 + 1 + 1 + ... + 1 = x$$
Or
$$2 = 1$$
Now, the mistake was that $x^2 = x + x + x + ... + x$ was not true $\forall x \in \mathbb{R}$ and therefore we cannot differentiate both sides and except them to be equal.
However, I learnt of discrete calculus and tried to take the discrete derivative of both sides and got:
$$2x + 1 = x$$ which is also not true for all $x \in \mathbb{N}$.
So, my questions are: Why doesn't the discrete derivative give correct results?
EDIT: I have already mentioned that the statement is not true for all reals and only for natural numbers. I think this makes some of the comments here redundant.