Ian Charlesworth

Suppose that $f : [a, b] \to \R$ is so that $\abs{f(t)} \leq M$ for all $t \in [a, b]$. Then $-M \leq L(f)$ and $U(f) \leq M$.

1. True.
2. True if $f$ is continuous.
3. True if $b-a \leq 1$.
4. False.

Suppose $f : \R \to \R$ is integrable on every compact interval, and define \[ F(t) = \int_0^t f(x)\,dx.\] If we wish to show that $F$ is continuous at $x_0$, which of the following quantities must we be able to force to be small?

1. $f(x) - f(x_0)$
2. $\int_{x_0}^x f(y)\,dy$
3. $\int_0^{x_0} f(x)\,dx$
4. $\int_{x_0}^x f(t) - f(x_0)\,dt$

Suppose $f : [0, 1] \to \R$ has the property that for every open subset of $[0, 1]$ contains a point at which $f$ is zero. Which of the following is true?

1. $f$ is integrable, and $\int_0^1 f(t)\,dt = 0$.
2. $f$ is not necessarily integrable, but if it is, $\int_0^1 f(t)\,dt = 0$.
3. $f$ is not necessarily integrable, and even if it is $\int_0^1 f(t)\,dt$ might not be zero, but it will be if $f$ is continuous.
4. None of the above.