Notes - Analysis II HT23, Intervals
Flashcards
Can you state what it means for a set $I$ to have the interval property?
\[\forall x,z \in I, x \le y\le z \implies y \in I\]
Suppose $f : I \to \mathbb{R}$ where $f$ is continuous and $I$ is an interval. What is always true about the image $f(I)$?
$f(I)$ is also an interval.
Suppose $f : [a,b] \to \mathbb{R}$ where $f$ is continuous. What is true about the image $f([a,b])$, and why do you need to be careful?
It is also a closed and bounded interval, but this is not true for all types of intervals.
If $f : I \to \mathbb R$ where $f$ is continuous and $I$ is an interval, what relates the injectivity of $f$ with its monotonicity?
\[f \text{ is injective} \iff f \text{ strictly monotonic}\]