(March 7, 2018 at 1:51 am)robvalue Wrote: An interesting feature is that limit of f(x) as x approaches t isn't always equal to f(t). In other words, the function can be converging on a value as you get arbitrarily close to a value of x, but it doesn't actually matter what happens exactly at the value, because you never get there. You can sometimes also get different results if you approach the limit from above or below.
For most functions people are familiar with however, the limit will be equal to the function at that point. These are continuous functions, which basically means there is no "jump" in their graph from one point to another.
Here is an example of a non-continuous function:
f(x) = x^2 for x not equal to 3
f(3) = 5
So this is a standard x^2 graph, except the value at 3 is 5 instead of the usual 9. There is a discontinuity there, a "jump". However, the limit of f(x) as x approaches 3 is 9, not 5.
Step functions are a good example of this. Tax rates are a good example: they suddenly jump at certain incomes (well, supposedly).
