Define arcwise(=path) connectedness of a set in a metric space. state a relation between arcwise connectedness and connectedness of a set.

See definitions and relation below

Step-by-step explanation:

Given points x and y of a certain set S in a metric space, a path from x to y is a continuous map  f:[a,b]-->S of some closed interval [a,b] in the real line into S, such that

f(a)=x and f(b)=y

In this case, we can also say that the points x and y are joined by a path or arc.

A set S in metric space is said to be path connected or arcwise connected if every pair of points x, y of S  can be joined by a path.

The relation between arcwise connectedness and connectedness of a set is that every arcwise connected set is also connected, but the converse does not hold; not every connected space is also path connected.

As an example, consider the unit square [0,1]X[0,1] with the dictionary order topology.

It can be proved that this space is connected but not path connected.

two data sets with the same median will not have the same varaibility

step-by-step explanation:

the graph in the attached figure

step-by-step explanation:

we have

using a graphing tool

the graph in the attached figure

the roots are ( x-intercepts) x=-0.5,x=1,x=3

the y-intercept is the point (0,3)