Introduction to Functions, Continuity of functions and Section Function

To understand the Continuity of Sections, let's first look at the meaning of functions and its continuity.

Functions:

Functions are something which takes an input from the set of points known as domain and gives exactly one output from that point. The set of all the points in the output is called range. In other words, each point in the domain map exactly to one unique point. Here, the output is dependent on what input we give to the functions. Let's see an example below.

If the function $f(x)$ is equal to the square of $x$, then
$$ f(x) = x^2 $$
Here, if we input different value of $x$ into the function $f(x)$, then we should get one unique value for that input. If $x=3$, then $f(x)=9$, and when $x=(-4)$, then $f(x)=16$. We can also draw the representation of funtions in graph. For the above example, the graph of the functions is as shown below.

Figure 1: Example of a function $f(x) = x^2$

Continuity of Functions:

If we look back at the definition of a continuous function in Calculus, then these three arguments should be true to a function to be continuous at point $a$:

1. The function value at $a$ exists, i.e. $f(a)$ exists. 
2. $\lim_{x\to a} f(x) $ exists.
3. $\lim_{x\to a}f(x) = f(a) $

We can also look at this concept from different perspective. 

Let's say that we have a function $f(x)$. For that function to be continuous from any range $(f(a)- \varepsilon, f(a)+ \varepsilon)$, there exists an interval $(a- \delta, a+\delta)$ such that for all $x \in (a- \delta, a+\delta) $, there exists $f(x) \in (f(a)- \varepsilon, f(a) + \varepsilon)$. If this scenerio is true, then the function $f(x)$ is continuous at $a$. If this is true for all the points in $f(x)$, then the whole function is continuous. 

For example, Figure 2 shows that the function is continuous while Figure 3 shows a discontinuous function at point $a$. 

Figure 2: A continuous function $f(x)$

Figure 3: A discontinuous function $f(x)$

Continuity of the Section Function:

If we recall from one of our last posts, then the definition of Section function is as follows:

$$s: X \times X \to PX $$ 

We also said that this is one of the most important elements in the field of Topological Robotics as it gives us the path between two points. 

Here, we will use the same method that we used to find the continuity of function to find the continuity of section. 

Let's take a neighborhood of a point $(x,y)$ on the Cartesian product and assume its radius as $\delta$. Let's take another point $(x',y')$ in the neighborhood of $(x,y)$. Using Section function, we get a path between $x$ and $y$, and another path between $x'$ and $y'$. If these two paths are closer to each other, then the section is continuous. Figure 4 is the example of a continuous section, while Figure 5 is an example of a discontinuous section. 

Figure 4: An example for continuous section

Figure 5: An example for discontinuous section

Posted by Chintan Patel.