by induction x+x+x+x....(x times)=x^2
Take derivatives of both sides of last statement:
1+1+1+1....(xtimes)=2X, but 1+1+1+1....x times equals x
therefore X=2X for all X, or specifically,
X + x + x ... (x times) is but an odd way to write "x times x", usually shortened x * x or x^2.
Y = x^2 is a continuous function.
Sun 14 Jun, 2020 05:42 pm
we are dealing with integers on the left side of the equation... I agree f(x)=x^2 is continuous and differentiable.....but consider g(x)= x+x+x+x..............x times. We know g(x)= x^2 is true if x is an integer....is it true if x=, say, pi ? is it true if x = -4.8 ?
For those that remember the definition of a derivative (lim as h goes to zero of (g(x+h)-g(x))/h, try to take the derivative of g(x) and it will fall apart.
However, if you think g(x) is differentiable, then there is nothing wrong with the proof and indeed, 7 does in fact equal 14