(34.4) Let be the function,
Determine the function .
Sketch . Where is continuous?
Where is differentiable? Calculate at the points of differentiability.
By the FTC II, is continuous everywhere.
By the FTC II, is differentiable everywhere that is continuous, hence at all except possibly and .
We can prove that is not differentiable at and by showing that the left-hand and right-hand difference quotient limits disagree.
(34.6) Let be a continuous function on and define
Show that is differentiable on and compute .
Let . Then is differentiable and by the FTC II as is continuous. Notice that . So by the chain rule, is differentiable and .
(34.11) Suppose is a continuous function on . Show that if , then for all .
Proof: We have that and is continuous as is, so by Theorem 33.4(ii) . By factoring, this is only possible if .
(34.12) Show that if is a continuous real-valued function on satisfying for every continuous function on , then for all .
Proof. As the statement is true for all , it is true for . Then this is true by question (3).
For this problem, you may use the results of questions (3) and (4) freely. Let be the set of all continuous functions on the interval . Define a function
Let . Show that is an inner product. That is, show each of:
if and only if .
Let . Show that if for all , then . In other words, you are showing that the only function orthogonal to all other functions with this inner product is the zero function.