arc wise connected set

Question

I am having confusion in understanding what is arc wise connected set.The definition is a set $S$ is arc wise connected if for any pair of point a,b we can define a continuous function $f$ from $[0,1]$ into S such that $f(0)=a$ and $f(1)=b$.I am not getting inuition of this definition.I cant understand why a set can not be arc wise connected.

for ex:the set $[-\infty,0)\cup(0,\infty] $ is not arc wise connected it means that we can have a continuous funtions $f$ defined as above,but i am not getting why.$f$ is defined at every point of the domain it just cant have the value $0$ which not part of S.Why does it mean f cant be coninuous.

can some one explain it with the example of a set which is not arc wise connected. I also want an example of a set which is connected but not arc wise connected as i am also having confusion in understanding difference of the 2.I am reading all this as basics for optimizaton so i dont know much about topology.I am following the book on optimization by kenneth lange which have defined this terms without using topology so please if some one

Answer 1

To understand it think it in $\Bbb R,\Bbb R^2,\Bbb R^3$. A subset $A$ of these spaces is arc wise connected iff for every $a,b\in A$ you can get a pencil and draw a curve from $a$ to $b$ without lifting the pencil up. Draw a continuous curve in other words.

In $(-\infty,0)\cup (0,+\infty)$ the pencil will be lifted at $0$ and you cannot do anything to stop it:P

Answer 2

The point is that the image of the arc must lie completely within $S$. Otherwise, every set that can be embedded in the plane (or in $\mathbb R^n$, for that matter), including two distinct points, would be arcwise connected because you could just use the segment joining them.

In the example you gave (extended real line with origin omitted), any continuous image of $[0,1]$ must lie completely in either the left side or the right side (remembering that the omitted point isn't allowed to be part of the image); that's because otherwise the inverse images of each of the two sides would be a pair of nonempty disjoint sets open in $[0,1]$ whose union was all of $[0,1]$, contrary to the connectedness of $[0,1]$, right?

The classic example of a space that is connected but not arcwise connected is the so-called "$\sin \frac1x$ continuum", consisting of the graph of $y=\sin\frac1x$ for $x\in (0,1]$ together with the vertical segment joining the points $\pm 1$ on the $y$-axis. The curve oscillates closer and closer to the segment along its entire length, so every point on the segment is a limit point of the curve, hence part of the same component as the curve. But there is no path within the whole set joining any point on the curve to any point on the segment.