Am I correct to say that if some function is discontinuous amid some range that it is defined over then we can make it continuous through the restriction of a finite interval that we choose?
As long as the choice of interval for the domain does not include the discontinuity. How does that help you?
Could you present an example on just how an integral fails over an interval that contains a point of discontinuity? From all the books I know of, I have never ever seen an example of the kind that demonstrates this anomaly in calculus. I've always been told about it but I've never seen an example to witness what happens when a point of discontinuity is involved.
You need to understand that there is a difference between a continuous function and a piecewise continuous function. Also, there's a difference between a continuous function and a continuously-defined function.
@WBahn I'd imagine that the difference between a continuous function and a piecewise continuous function is that the former is defined over an infinite domain from - infinity to + infinity such as the derivative of a periodic function like y=sin(x), for example. Whereas the latter is restricted in which it is continuous such as In(x), for example. For the remainder of your question, I'm not sure. Edit: Could you look at post #3. I've kind of merged two threads into one.
That was a quick gear shift. We went from talking about continuous functions to doing integrals. Functions don't have to be continuous to be integrable. They do have to be continuous to be differentiable.
Integrals don't necessarily fail when there is a point of dicontinuity in the interval of integration. Examples are the step (heaviside) function and the dirac delta function. In fact the particular value of the function chosen at the discontinuity is irrelevant to the evaluation of the integral almost surely. http://en.wikipedia.org/wiki/Almost_surely
A continuous function (and there is always a defined domain, though it may be implied) is essentially one in which you can draw the function without ever lifting your pen. Another way of putting it more mathematically is that for every point in the domain, x , the difference between f(x) and f(x+ε) goes to zero as ε goes to zero. A piecewise continuous function is allowed to be discontinuous at a finite set of points and, does not even need to be defined at those points of discontinuity. A continuously defined function has a finite value at every point in the domain. So a piecewise continuous function need not necessarily be continuously defined. As Papabravo says, a function that is piecewise continuous (and thus may not even be defined at some finite number of points) is still integratable. But if it is defined such that there are an infinite number discontinuities then this may not be the case.