On 9/29/2013 6:29 PM, FredJeffries wrote: > On Saturday, September 28, 2013 1:15:47 PM UTC-7, Hetware wrote: >> >> So the answer is consensus among mathematicians holds that F(t) = (t^2 - >> 9)/(t - 3) is undefined at t=3? Perhaps what I should have said at the >> outset is something along the lines of: on any given day, if I'm setting >> up an equation in physics, and produce an expression such as F(t) = (t^2 >> - 9)/(t - 3), I treat it as t+3, and do not expect any adverse >> consequence from doing so. > > But you are not setting up an equation in physics. You are > attempting to learn calculus. >
No. I'm reading the chapter for the nth time. I already know calculus. I am perfectly capable of accepting something as true and follow arguments based upon that predicate.
> Professor Thomas has a very good reason for showing you this > trivial-seeming example of a function with what is known > as a "removable singularity". If you are patient and continue > to study his book you will find his reasons. Here's a hint: > the function sin(x)/x has the same feature at x = 0.
I'm not sure how the first prepares me for the second.
>> If I conceive of mathematics as an exercise in defining and manipulating >> symbols, it seems that declaring constructs such as F(t) = (t^2 - 9)/(t >> - 3) to be undefined at t=3 is arbitrary. The fact that there is an >> obvious candidate for a value of F(t) at t=3 tells me that accepting >> that candidate as the value at t=3 does not contradict the definition of >> a single valued function of one variable. > > Of course it does not contradict the definition of a single valued > function of one variable. It contradicts the definition of one > particular function, namely Professor Thomas's F(t). The fact that > this all seems totally arbitrary to you is just because you > don't know the whole story yet. >
A function is a mapping from elements of a range to elements of a domain. Often the image is required to be single valued, but Thomas does not stipulate that requirement. If I can deterministically interpret a formal expression as such a mapping, then my interpretation of that formal expression satisfies the definition of a function.