Chapter 1 – Quiz 2 – E600 Mathematics

1. Which of the following statements about set properties are true? (Multiple choice)

The interval $I = (2,4)$ is an open ball in $\mathbb R$ (using the natural metric).

If $A$ is a bounded set, $A$ can not be a closed set.

The interval $I = [0,1]$ is an open ball in $\mathbb R$ (using the natural metric).

If $A$ and $B$ are open sets, then $A\cup B$ is an open set.

If $A$ is a closed set, $A$ can not be an open set.

If $A$ is a compact set in $\mathbb R^n$ , then $A$ is also bounded.

The empty set is neither open nor closed.

2. Consider the function $d:\mathbb R \mapsto \mathbb R, (x,y) \mapsto |x+y|$ . Which property of the metric concept is violated? (Single choice)

Non-negativity.

Symmetry.

Triangle inequality.

None; $d$ is a metric.

3. Which statements about metric and norm are true? (Multiple choice)

Metrics can not be used to express distances mathematically.

The norm of a vector $x$ , $\|x\|$ , can be thought of as the distance to the origin (the zero vector).

If $\|\cdot\|$ is a norm on a set $X$ , then $d:X\mapsto\mathbb R, x\mapsto \|x\|$ is a metric on $X$ .

Any norm induces a metric as the norm of the vector difference.

If $\|\cdot\|$ is a norm on a set $X$ , then $d:X\times X\mapsto\mathbb R, (x,y)\mapsto \|x-y\|$ is a metric on $X$ .

Any norm is also a metric.

4. What is true about the general continuity concept, that is, for continuity of a general function $f:X\mapsto Y$ mapping between vector spaces? (Single choice)

If $\lim_{n\to\infty} x_n = x_0$ and $\lim_{n\to\infty} f(x_n)\neq f(x_0)$ , then $f$ is not continuous at $x_0$ .

It is not possible to generalize the $\varepsilon-\delta$ approach to the definition of continuity from the univariate case to more general functions.

A continuous function can have some points at which it is not continuous.

It is possible that $f$ is continuous at $x_0$ even if $\lim_{x\to x_0}f(x) = f_0$ with $f_0\neq f(x_0)$ .

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