tightgas@nmtvax.UUCP (09/12/83)
Larry Cipriani submitted a 2 = 1 'proof' that began with x^2 = x + x + ... + x (x times). The number of terms on the right-hand side is also a function of x. Therefore, IF the right-hand side can be differentiated with respect to x (however, see below **), it is subject to differentiation just as though the equation were written as x^2 = x*x Thus, 2*x = 1*x + x*1 2*x = 2*x 2 = 2 ---------- ** Presumably, the number of times x is written on the right-hand side is an integer. Since this is x times, it appears that x can only have integer values. Therefore, x is not continuous. I think that x has to be continuous in order to differentiate with respect to x. Neal Kilmer New Mexico Tech
rab@cdcvax.UUCP (Roger Bielefeld) (09/13/83)
Actually, in a nutshell, differentiability implies continuity but continuity does not imply differentiability. An example of a function that is continuous but not differentiable is f(x) = |x| on any open interval containing zero. Of course, we're talking about real functions of real variables here.
prodeng@fluke.UUCP (Jim Hirning) (09/14/83)
Regarding differentiating both sides of x^2 = x + x + ... x (x times). It is true that x must be integer; it is also true that since x is not continuous, this function cannot be differentiated. To differentiate a function, you must be able to take a limit, thus continuity is required. I also *suspect* that since the number of x's on the right hand side is indefinite, the purported equation really is illegal to start with. (At least, an induction argument should be supplied). Debbie Smit fluke!prodeng