MATH 3912 - Assignment 2
1. Recall that a function f:D7→ Ris bounded if there exists a constant Msuch that |f(x)| ≤ Mfor
all x∈D. For our discussion assume that Dis an interval such as (a, b), [a, b], (−∞, b), (a, ∞), or
corresponding half-open intervals on the real line.
Find a function that is not bounded. Is it true that every continuous function is bounded? If not,
find conditions on the domain that ensure that continuous functions are bounded. Are all differentiable
functions bounded?
2. For what values of aand bare the functions f(x) = axbbounded on [0,1]?
3. For what values of aand bare the functions f(x) = 1
x2+ax+bbounded on [−1,1].
4. Consider the function f(x) = 1/x2. Is the function:
(a) Continuous on the interval [0,1]?
(b) Continuous on the interval (0,1)?
(c) Continuous on the interval [1,2]?
(d) Continuous on the interval [1,∞)?
(e) Uniformly continous on the interval [0,1]?
(f) Uniformly continous on the interval (0,1)?
(g) Uniformly continuous on the interval [1,2]?
(h) Uniformly continuous on the interval [1,∞)?
5. Show that the function f(x) = e−xis uniformly continuous over the interval [0,∞). What about on the
interval (−∞,0) (explain by means of a picture)?
6. Recall that a function fis Lp[a, b] if Rb
a|f(x)|pdx exists.
(a) Is the function f(x) = 1
x∈L1[0,1]?
(b) Is the function f(x) = 1
x1/2∈L1[0,1]?
(c) Is the function f(x) = 1
x1/2∈L2[0,1]?
(d) For a fixed 0 ≤α < 1 find psuch that f(x) = 1
xα∈Lp[0,1]?