Thursday, June 14, 2012

Limits and Continuity

In this post, I am going to explain the concept of continuity in calculus in a bit more detail than when I touched on the subject in my previous post that explained one-sided limits.  I will have even more to say about the concept of continuity when I begin my series on derivatives soon, as derivatives can quite easily provide you with an assessment of the continuity of a graph.

As we've seen, limits calculus is a branch of mathematics that describes what a graph is doing as you approach a point on the graph and get infinitely close to it without ever actually reaching it.  It may be thought of as a way of predicting what a graph will do at a point, based on what the graph is doing in the general vicinity of that point.

For instance, if you imagine an upwards facing parabola, if you begin to trace the curve with your finger, moving towards the parabola's apex, you find that you seem to be approaching a predetermined point.  This holds true, in this example, if you are moving towards the apex from either the left side or from the right side.  Based on the curvature of the parabola, you can predict what the point will be as you approach it.  Also, you can see that the point actually is exactly what the line would predict it to be.  Since this is a simple example and very easy to visualize, you can intuitively understand just by looking at it that the graph is continuous.  There are no holes or anything weird going on along the length of the curve.  It is a smooth, continuous line.

However, if you consider a graph like the one I showed you when I talked about one-sided limits in calculus, you can very easily see, without any form of analysis at all, that the graph is discontinuous. You can't draw that line without lifting your pen off the paper.  In fact, this is a very crude but simple way of checking to see if a graph is continuous or discontinuous.

In calculus, limits can explain more than just what point a graph seems to be going towards.  Limits can be used to tell us about the continuity of a graph.  In fact, limits and continuity are very important parts of graph analysis.  If you want to know if a graph is continuous at a certain point, you merely need to look at the one-sided limits on both sides of that point, as well as the point itself.  If you look at the parabola above, and want to know if it's continuous at its apex, you consider the limit as you approach from the left, and also the limit as you approach from the right, and in this case you see that they both approach the apex.  There apex is also on the curve.  If the two one-sided limits equal the same number, and that number also evaluates to the same value at that point, you can conclude that the graph is continuous at that particular point.  For similar reasons, you can see that the second graph above is a discontinuous graph at x=5.

I will leave this example for you to draw and consider: imagine a normal parabola, as above, but it's apex is undefined (a hole in the graph).  The one-sided limits from both sides approach the same value, but at the apex the graph is undefined, which obviously doesn't equal the one-sided limits result.  Therefore, the graph would be discontinuous.

It should be fairly obvious, but I'd better state it here anyways, that this doesn't say anything about the continuity of the graph at any other point.  A graph could very well be continuous in one place and discontinuous somewhere else.  When examining the one-sided limits and the value at that point, you are only evaluating the continuity of the graph AT THAT POINT.  You are not evaluating the continuity of the graph as a whole.  It's safe to say, however, that if you have a point on a graph that is discontinuous, then you have a discontinuous graph overall.  To have a continuous function, you have to have continuity at every point on the graph.  That is to say, f(x) equals both of the one-sided limits for every value of x.

Most graphs that you will come across will be continuous, though you will need to pay special attention to functions that are piecewise as they may or may not be.  By definition, all polynomials are continuous.  So are rational functions (fractional expressions) except when the denominator is 0.  This is because a rational expression with 0 in the denominator is undefined, so it has no point on the graph.  So, obviously, it isn't continuous where there is no point!

So, to quickly summarize, to evaluate the continuity of a graph of f(x) at a specific point, you need to determine the one-sided limits as you approach that point from either side, and you also have to evaluate f(x) at that specific point.  If f(x) equals both one-sided limits, you have proven that the graph is continuous at that point.

I hope that this post makes sense and explains limits and continuity calculus for you.  If you need something to be explained better, or maybe a different example, please leave me a comment and I will add some more information.  Also, please +1 me below this if you found it useful, as it really helps me!  Thanks for reading.

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