ASSIGNMENT 2·9 for SECTION 001
There are two parts to this assignment. The first part is online at www.mathxl.com. The second part consists
of the questions on this page. Your solutions must be typeset, preferably using LATEX. You are expected to
provide complete arguments and full justifications. Your paper must be stapled, with your name and student
number at the top of each page.
In mathematics, a pathological function is a function with one or more counterintuitive properties. For
example, the Dirichlet function from previous assignments and workshops is a pathological function.
In this assignment, you will examine a pathological function which illustrates the need to define the property
“increasing” on an interval, not a point. In most “normal” cases, if f 0 (c) = 0, then f turns out to be
increasing on an interval containing 0; but not in this case. The function in question is
if x 6= 0
x + 2x2 sin x1
.
f (x) =
0
if x = 0
1. Prove that f is continuous on the interval (−∞, ∞).
2. Prove that f 0 (0) = 1. (Hint: use the limit definition of derivative.)
3. Explain why any interval containing 0 must also contain points x where f 0 (x) is negative. (This shows
that f is not increasing on any such interval, despite the fact that f 0 (0) = 1.)