. Hint:It might help to write
this in terms of the variables
(s,t)
=
(x− 1,y− 2)
.
Suppose limx→0f
(x,0)
= 0 = limy→0f
(0,y)
. Does it follow that
lim f (x,y) = 0?
(x,y)→(0,0)
Prove or give counter example.
f : D ⊆ ℝp→ ℝq is Lipschitz continuous or just Lipschitz for short if there exists a constant K such
that
|f (x )− f (y)| ≤ K |x − y|
for all x,y∈ D. Show every Lipschitz function is uniformly continuous which means that given ε > 0
there exists δ > 0 independent of x such that if
|x − y |
< δ, then
|f (x)− f (y)|
< ε.
If f is uniformly continuous, does it follow that
|f|
is also uniformly continuous? If
|f|
is uniformly
continuous does it follow that f is uniformly continuous? Answer the same questions with “uniformly
continuous” replaced with “continuous”. Explain why.
Let f be defined on the positive integers. Thus D
(f)
= ℕ. Show that f is automatically continuous at
every point of D
(f)
. Is it also uniformly continuous? What does this mean about the concept of
continuous functions being those which can be graphed without taking the pencil off the
paper?