Make a list of all definitions in the section (a few words each is fine). Summarize any lengthy definitions introduced in this section in your own words.
The intuitive idea is that in a small neighborhood of x=a, the function f takes on values close to f(a). A function can be discontinuous from the left only,
or from the right only:
(which must exist for the removable discontinuity).
does not exist, because
There is a ``jump'' in the graph.
Make a list of all theorems (lemmas, corollaries) in the section (a few words each is fine). Summarize each one introduced in this section in your own words.
Here again, all the properties we'd like continuous functions to have are indeed satisfied: the sum of continuous functions is continuous, etc.
This is one of the most important theorems in this section.
Note that this is not the same as f(g(a)), which may not exist.
is continuous at a, and
``A continuous function of a continuous function is continuous.''
(note: this only holds if ).
This theorem effectively says that in crossing a street you have to pass every point in the street: you can't get from one side to the other without going through the middle.
Make a note of any especially useful properties, tricks, hints, or other materials.
Summarize the section in two or three sentences.
Continuity is defined and discussed in detail, along with varieties of discontinuities (jump, infinite, removable). Once again the theorems suggest that properties we'd like continuity to possess are true: e.g., the sum of two continuous functions is continuous; the composition of continuous functions is continuous. Entire classes of functions are continuous on their domains (polynomials, rational functions, trigonometric functions, root functions), so limits for these are easy to compute.
Problems:
Assignment: p. 112-114, #4-8, 13, 38, 40, 45, 48, 59
In class: #3, 10, 12, 16, 19, 33, 36, 44