fmiren said:
Thank you. How does it apply to this problem? Is the function discontinuous at c, e, g, and l points? And what it implies?
This has to do with picky little details about the definition of continuity. It has little to do with the problem posed in your original post.
Specifically, we are getting at the question of whether a function is continuous at its endpoints.
Intuitively, for a function to be continuous at a point, the graph of that function must not have a break at that point -- you should be able to draw the graph without lifting your pencil from the paper. You are also not allowed to have any perfectly vertical lines on the graph, but that's part of what it means to have a function in the first place.
When you get into a math course where continuity is more carefully defined you find a definition along the lines of:
A function f() is continuous at a point l iff
1. l is in the domain of f
2. for all epsilon > 0 there is a delta > 0 such that for all x
in the domain of f such that | x - l | < delta, | f(x) - f(l) | < epsilon
Paraphrased, that says if you look in a neighborhood that is close to l, function values in that neighborhood will be close to f(l). No matter how close you need the function values to be, you can always find a neighborhood that is small enough to be sure that the function values in that neighborhood will all be that close.
The point here is that it does not matter for the purposes of continuity whether you are at the edge of the function's domain or not.
x values that do not fall within the function's domain do not count against continuity.
Wiki points this out:
"It follows from this definition that a function
f is automatically continuous at every
isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous."
