Continuity
due by Monday, Sep 13, 2021
This assignment is based on material in lectures 7 and 8.
-
Let
$f: X\to Y$
be a function and$A, B\subset X$
. Prove or disprove the following.$f(A\cup B)\subset f(A)\cup f(B)$
.$f(A\cup B)\supset f(A)\cup f(B)$
.$f(A\cap B)\subset f(A)\cap f(B)$
.$f(A\cap B)\supset f(A)\cap f(B)$
.
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Let
$(X, d_X)$
and$(Y, d_Y)$
be metric spaces. A map$f: X\to Y$
is said to be Lipschitz if there exists a constant$K> 0$
such that$$\forall p, q\in X,\ d_Y(f(p), f(q)) \leq K d_X(p, q).$$
Prove or disprove the following.- Every Lipschitz map is continuous.
- Every continuous map is Lipschitz.
-
Let
$Y$
be a topological space such that for all topological spaces$X$
, every map$f: X\to Y$
is continuous. Prove that$Y$
has the indiscrete topology. -
Suppose $f: X\to Y$ is a map between topological spaces and $A\subset X$ is a subspace. Prove or disprove the following.
- If the restriction $f\vert_A: A \to Y$ is continuous, then for all $x\in A$, $f$ is continuous at $x$.
- If, for all $x\in A$, $f$ is continuous at $x$, the restriction $f\vert_A: A \to Y$ is continuous.
-
Let $X$ be a space with the discrete topology. Prove or disprove the following.
- If $x\in X$, then any neighbourhood basis of $x$ contains the singleton set ${x}$.
- Every point has a neighbourhood basis with just one open set.
-
Let
$f, g: X\to \mathbb{R}$
be continuous functions from a topological space to the reals. Prove or disprove that the following must be continuous?$\min(f, g)$
.$\max(f, g)$
.$\vert f\vert - \vert g\vert$
.