Question is :
Suppose $f: [0,1]\rightarrow (0,1)$ is Continuous then which of the following is NOT true..
- $F\subseteq[0,1]$ is closed set implies $f(F)$ is closed in $\mathbb{R}$
- If $f(0)<f(1)$ then $f([0,1])$ must be equal to $[f(0),f(1)]$
- There must exist $x\in(0,1)$ such that $f(x)=x$
- $f([0,1])\neq (0,1)$
Continuous map need not map closed sets to closed sets..
So, first option is not true…
Continuous maps takes connected sets to connected sets …
So $f([0,1])$ must be connected and it is equal to $[f(0),f(1)]$.. So, Second option is true..
Continuous maps takes compact sets to compact sets…
So, $f([0,1])\neq (0,1)$ and so fourth option is true…
I guess third option is also false though I can not think of any example..
Continuous map from compact set to itself has a fixed point.
But how do i conclude that this would imply third option is true/false.
Please help me to clear this…
Thank you.
Best Answer
Constructing the counterexample to (2) is easy. Draw a picture! Can you draw a squiggly line from $f(0)$ to $f(1)$ that goes below $f(0)$ say? That's your counterexample!