Continuity and Differentiability
Two big ideas to address here. As a result this page is long.
The graph of a continuous function has no holes, jumps, or gaps. Think of a continuous function as one that you can graph without ever lifting your pencil. Such a function would be considered continuous over its entire domain. For an example, look at functions f and g below.
On the other hand, functions p and q are not continuous over their entire domains. Both functions are discontinuous at x = 3. We should be careful to note that a function that is not continuous everywhere can be continuous on sub-intervals of the domain. For example, function q is continuous on the interval [-1, 3].
You must memorize the following test for determining if a function is continuous at a specific location. To determine if a function, f(x), is continuous at x = c, the following must be true.
Really, part 3 is is what we need to look at since parts 1 and 2 are already built into part 3. However, on an AP exam free response question asking about continuity at a specific location, it is important that you clearly demonstrate all three parts listed above. Here is another way to think of these three steps.
Below are two full lessons that detail continuity.
Two shorter videos with examples.
DIFFERENTIABILITY
A function is considered differentiable if you can find its derivative. Just as it was with continuity, a function can be differentiable over its entire domain or over sub-intervals of the domain. Remember that a derivative represents the slope of a tangent line. Therefore, to ask if a function is differentiable is to ask if you can draw lines tangent to the function.
We can not draw tangent lines at cusps, holes, jumps, or endpoints. Therefore, a function is not differentiable at any cusps, holes, jumps, or endpoints. Understanding this leads to the following conclusion:
Differentiability implies Continuity
but
Continuity does NOT imply Differentiability
but
Continuity does NOT imply Differentiability
If a function is continuous on an interval, you can draw no conclusion as to whether or not it is differentiable on that interval. There may be cusps. But if a function is differentiable on an interval, it must be continuous on that interval.
For more review...
HOMEWORK Check out this video for help on problem #4. And here is an example of how to tackle problem #11.