252 CHAPTER 5 Series and Differential Equations
(b) limx→af(x) = L,
(c) gis defined in a punctured neighborhood ofL, and
(d) limx→Lg(x) = M.
Show that limx→ag(f(x)) = M.- Show that ifa≥0, then limx→a
√
x=√
a.- Show that ifa∈R, then limx→a
√
1 +x^2 =√
1 +a^2.- Prove that the sequence 1, 0 , 1 , 0 , ...does not have a limit.
- Let f :D→ R be a real-valued function, whereD(the domain)
is some subset of R. Prove thatf is continuous at a∈Dif and
only if for every sequence a 1 , a 2 , ..., which converges to a, then
nlim→∞f(an) =f(a). - Here we revisit Exercises 11 and 12 on page 239.
(a) Let f :R → R be a differentiable homomorphism, i.e.,f is
differentiable and satisfiesf(x+y) =f(x) +f(y) for allx, y∈
R. Prove that there existsc∈Rsuch thatf(x) =cx, x∈R.
(This is easy!)
(b) Let f : R → R be a continuous homomorphism, and prove
that the same conclusion of part (a) holds. (Hint: you want to
prove that for all a, x∈ R, f(ax) =af(x). This guarantees
thatf(x) =xf(1).)^3 - Definef(x) =
...
x+√
x+√
x+···, x≥0. Clearlyf(x) = 0. Is
f continuous atx= 0? (Hint: note that if we sety=f(x), then
y=√
x+y, and soy^2 =x+y. Using the quadratic formula you
can solve foryin terms ofx.)- Recall Euler’sφ-functionφ, given on page 63. Define the setAof
real numbers
A =
φ(n)
n|n∈N
.(^3) It may surprise you to learn that “most” homomorphismsR→Rarenotcontinuous. However,
the discontinuous homomorphisms are essentially impossible to write down. (Their existence involves
a bit of advanced mathematics: using a “Hamel basis” for the real numbers, one can easily write
down discontinuous homomorphismsR→R.)