On 10/6/2013 11:45 AM, Peter Percival wrote: > Hetware wrote: > >> OK. I turned this around and can see the problem. I was assuming f(t)= >> (t^2-9)/(t-3) is continuous, which is not justified from the context. At >> t=3, x can be anything in t^2 - 9 = x (t - 3). > > If f is a function with domain the most inclusive set of reals possible, > then f is continuous. >
If I say f(t) is continuous for all real numbers, and define f(t)=(t^2-9)/(t-3) for all t!=3, then limit[f(t),t->3]=6. So f(3)=6, by the definition of continuity. A function thus defined is identical to g(t)=t+3.
The assertion that the f(t) is continuous determines the value of f(t) at t=3 without the need for further elaboration. So, I concede that I was wrong in believing that f(t) defined as f(t)=(t^2-9)/(t-3) is identical with g(t)=t+3. The continuous function defined as f(t)=(t^2-9)/(t-3) everywhere the rhs is defined is identical with g(t)=t+3.