Chapter 1

Prove that if \(f:[a, b] \rightarrow \mathbb{R}\) is a regulated function and \(F:[a, b] \rightarrow \mathbb{R}\) is defined by \(F(x)=\int_{a}^{x} f(t) d t\), then \(F\) is continuous.

Show that the continuous function \(f(x)=1 / x\) on the open interval \((0,1)\) is not regulated, i.e., it cannot be uniformly approximated by step functions.

Prove that the collection of all step functions on a closed interval \([a, b]\) is a vector space of functions which contains the constant functions.

Prove that the absolute value of a Riemann integrable function is Riemann integrable.

Give an example of a bounded continuous function on the open interval \((0,1)\) which is not regulated.

Prove that if \(x_{0}=a

Suppose \(f\) and \(g\) are Riemann integrable functions defined on \([a, b] .\) Prove that if \(h(x)=\max \\{f(x), g(x)\\}\), then \(h\) is Riemann integrable. This generalizes to the max of a finite set of functions, but not of infinitely many. Show there exists a family \(\left\\{f_{n}\right\\}_{n \in \mathbb{N}}\) of step functions such that for each \(n\) and each \(x \in[a, b]\) the value of \(f_{n}(x)\) is either 0 or 1 and yet the function defined by \(g(x)=\max \left\\{f_{n}(x)\right\\}_{n \in \mathbb{N}}\) is not Riemann integrable.

Let \(f:[a, b] \rightarrow \mathbb{C}\) be a complex-valued function and suppose its real and imaginary parts, \(u(x)=R(f(x))\) and \(v(x)=\Im(f(x))\), are both continuous. We can then define the derivative (if it exists) by \(d f / d x=d u / d x+i d v / x\) and the integral by $$ \int_{a}^{b} f(x) d x=\int_{a}^{b} u(x) d x+i \int_{a}^{b} v(x) d x $$ (a) Prove that if \(F:[a, b] \rightarrow \mathbb{C}\) has a continuous derivative \(f(x)\) then $$ \int_{a}^{b} f(x) d x=F(b)-F(a) $$ i.e., the fundamental theorem of calculus holds. (b) Prove that, if \(c \in \mathbb{C}\) and \(F(x)=e^{c x}\) for \(x \in[a, b]\), then \(d F / d x=c e^{c x} .\) Hint: Use Euler's formula: $$ e^{i \theta}=\cos \theta+i \sin \theta $$ for all \(\theta \in \mathbb{R}\). (c) Prove that, if \(c \in \mathbb{C}\) is not 0 , then $$ \int_{a}^{b} e^{c x} d x=\frac{e^{c b}-e^{c a}}{c} $$

Give an example of a sequence of step functions which converges uniformly to \(f(x)=x\) on \([0,1]\).

Prove that if \(f\) and \(g\) are bounded Riemann integrable functions on an interval \([a, b]\), then so is \(f g .\) In particular, if \(r \in \mathbb{R}\), then \(r f\) is a bounded Riemann integrable function on \([a, b]\)

Prove that the collection of all regulated functions on a closed interval \(I\) is a vector space which contains the constant functions.

( \(\star)\) Prove that \(f\) is a regulated function on \(I=[a, b]\) if and only if both of the limits $$ \lim _{x \rightarrow c+} f(x) \quad \text { and } \quad \lim _{x \rightarrow c-} f(x) $$ exist for every \(c \in(a, b) .\) (See section VII.6 of Dieudonné \([\mathbf{D}] .)\)