I haven't looked into it much, but this is something I've been aware of that I know I need to look into.
When I have a function $f(x)=\frac{x+1}{x+1}$, There is a discontinuity at $x=-1$, yet $\frac{x+1}{x+1}=1$ and has no discontinuity. It's like they're equal but not.
The qualities of the function are not preserved after the algebraic manipulation, so I can't strictly say that $\frac{x+1}{x+1}=1$.
This is an issue for me when understanding integrals. For instance, finding the definite integral of the quotient, if the discontinuity is within my limits, doesn't make sense. But after changing the quotient to a constant, it's possible: but I've found the area under a curve that wasn't complete. I've found a solution for an unanswerable, insensible question.
I hope I've made this clear. My question is, is this right? How do I come to terms with this?