- Let f= sinfor x ∈ (0,1] and let f= 0 . Show that f is Riemann integrable. Is f piecewise continuous?
- Show that if f is Riemann integrable on and= f everywhere except at finitely many points, thenis Riemann integrable and
- Suppose f is a decreasing function defined on . Show that
Now let δ

_{n}be a decreasing sequence of positive numbers converging to 0 and letso that

is an interval no longer thanδ_{m}. Also, these sets are decreasing. Argue there exists I in all of these intervals and that wheneveris sufficiently small,< ε. Thus f is Riemann integrable. - Give a version of the above problem to the case where f is increasing.
- Use the above two problems to show that if f : → ℝ is a bounded function and there are sub-intervals determined by the partition a = x
_{0}< x_{1}<< x_{n}= b such that onthe function f is either increasing or decreasing, then f is Riemann integrable. This shows that the only functions which are not Riemann integrable are very wriggly. Hint: Follow Corollary 6.1.12. - Show that if f is Riemann integrable on then so isand if f,g are Riemann integrable then so is αf + βg. Use to generalize the theorems in the section on properties of the integral. Hint: Recall the triangle inequality,≤.
- Generalize Theorem 6.2.8 to the case where f is only Riemann integrable on and continuous at t. Hint: It goes the same way.
- Show that if f is continuous on , then ∫
_{a}^{b}fdx = ffor some c ∈. Hint: Reduce to the mean value theorem for derivatives and use Theorem 6.2.8. - The first order linear initial value problem for the unknown function y is of the
form
where y

_{0}is given and p,qare given continuous functions. How do you find y in this problem? This will be discussed in this problem. Incidentally, this is the most important equation in differential equations and properly understood includes almost the entire typical undergraduate differential equations course. Let P≡∫_{0}^{x}pdt. Then multiply both sides by expand when you do, show that you obtainNow explain why, when you take an integral of both sides, you get

Then

- One of the most important inequalities in differential equations is Gronwall’s
inequality. You have
where t → u

is some continuous function usually nonnegative. Then you can conclude thatExplain why this is so. Hint: Let w

= ∫_{0}^{t}uds and write the inequality in terms of w and its derivatives. Then use the technique of the previous problem involving integrating factors. - A function f satisfies a Lipschitz condition if ≤ K. A standard initial value problem is to find a function of t denoted as y such that
where y

_{0}is a given value called an initial condition. Show that this initial value problem has a solution if and only if there is a solution to the integral equation(6.7) Hint: This is an application of theorems about continuity and the fundamental theorem of calculus.

- Letting f be Lipschitz continuous as in 6.7, use Gronwall’s inequality of Problem
10, to show there is at most one function y which is a solution to the integral
equation 6.7. Hint: If y,ŷ both work, explain why
Also give a continuous dependence theorem in the case that you have y,ŷ solutions to

respectively. Verify

e^{Kt}≥. - Give an example of piecewise continuous nonnegative functions f
_{n}defined onwhich converge pointwise to 0 but ∫_{0}^{1}f_{n}dx = 1 for all n. This will show how uniform convergence or something else in addition to pointwise convergence is needed to get a conclusion like that in Theorem 6.3.1. - Let F= ∫
_{x2}^{x3 }dt. Find F^{′}. - Let F= ∫
_{2}^{x}dt. Sketch a graph of F and explain why it looks the way it does. - Let a and b be positive numbers and consider the function
Show that F is a constant.

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