The function H(x) defined by
Find the conditions on that make
1. Continuous at
2. Differentiable at
1. In order to make continuous at , the function needs to satisfy the following conditions:
(1) is defined;
(2) exists; and
(3)
To satisfy (1), consider , we need is defined.
To satisfy (2), we need to make sure both and exist, also .
Here we have and .
So we need and .
To satisfy (3), realize that from above, if exists then . Also we have , hence .
In summary, in order to make continuous at , function has to satisfy .
2. Given that , so in order to make differentiable at , we need to show the limit exists. More specifically, we need to show the one-sided limits and exist and are equal. Can you pick up from here?
Roy