A Brief Note on Properties of Continuous Functions

In mathematics, a continuous function is a function that does not contain any discontinuities, which indicates that its value does not change in an unexpected way at any point. We can ensure arbitrarily small changes in a function by confining minor changes in its input to a sufficient number of minor changes. If a particular function is not continuous, it is referred to be discontinuous in this context. With another way of saying it, we can state that a function is continuous at a fixed point if we are able to draw a graph of that function around that point without raising the pen from the plane of the paper. 

Continuous functions definition: 

Mathematically, we may define the continuous function by utilising the limits that are shown in the following example:

Take, for example, an imaginary function on a subset of the real numbers, and consider c to be a point within the domain of f. f is continuous at c if and only if 

limx→c f(x) = f(c) 

It is possible to enlarge on the previous definition by saying that if the left-hand limit, right-hand limit, and the function’s value at x = c are all present and equal to each other, the function f is continuous at x = c. Once we have determined that the limits of the right hand and left-hand limits of the function at x = c are identical, we can conclude with confidence that the expected value equals the function’s limit at x = c. We may also rephrase the concept of continuity as “a function is continuous at x = c if the function’s definition is at x = c and the function’s value at x = c equals the limit of the function at x = c” in order to make it more precise. It is possible to claim that f is not continuous at c, in which case it is discontinuous at c, and this point is referred to as c as a point of discontinuity of the provided function.

The following section describes another method of defining the continuous function.

Continuity is defined as the presence of the function f at every point in the domain of the function. This can be explained in further depth using mathematical language as follows:

To be considered continuous, let us suppose that f is defined on the closed interval [a,b]. In order for f to be defined as such, it must be continuous at every point in [a,b], including the ends of the interval [a,b].

The continuity of f(x) at a given point 

limx→a⁺ f(x) = f(a) 

and continuity of f(x) at b means 

limx→b⁻ f(x) = f(b) 

If the function is defined only once, then it is continuous in that place; in other words, if the domain of the function is a singleton, then the function will be a continuous function.

We may derive three requirements for determining whether or not a function is continuous from the definitions provided above. They are as follows:

Consider the relationship between the function f(x) and the point x = a. 

1. In order for the function to be continuous at point x = a, it must be defined at that point.

2. The limit of the function f(x) should be defined at the point x = a, which is the origin of the graph.

3. f(a) is the value of the function f(x) at that point, which must be identical to the value of the limit of the function f(x) at the value of a. 

Boundedness theorem: 

Specifically, it states that if a function f(x) is continuous on a closed interval [a,b], then the function is also bounded on that interval: there exists a constant N such that the size (absolute value) of f(x) is at most N for all x in [a,b]. This is not necessarily true if f is only continuous on an open (or half-open) interval: for example, the function 1/x is continuous on the open interval (0,2018), but it is unbounded on the closed interval (0,2018). 

Extreme Value Theorem: 

On specific intervals, critical points are useful in determining the feasible maximum and lowest values of a function, which is an important application of critical points. According to specific conditions, the Extreme Value Theorem assures the existence of a maximum and a minimum value for a function. The following is stated in the document: 

• If a function f(x) is continuous on a closed interval [ a, b], then f(x) has both a maximum and minimum value on [ a, b]. 

First, it is necessary to prove that the function is continuous on the closed interval before proceeding with the application of the Extreme Value Theorem. The following step is to identify all critical points within the supplied interval and to evaluate the function at each of these critical points as well as at the beginning and ending points of the given interval. 

Intermediate Value Theorem: 

Suppose “f” is a continuous function over the closed interval [a, b], and its domain contains the values f(a) and f(b) at the interval’s ends, then the function takes any value between the values f(a) and f(b) at any point within the interval, according to the intermediate value theorem. This theorem is explained in two ways: first, as follows: 

Statement 1: 

If k is a value between f(a) and f(b), i.e.

either f(a) < k < f(b) or f(a) > k > f(b).

Then there exists at least a number c within a to b i.e. c ∈ (a, b) in such a way that f(c) = k. 

Statement 2: 

The set of images of function in interval [a, b], containing [f(a), f(b)] or [f(b), f(a)], i.e.

Either f([a, b]) ⊇ [f(a), f(b)] or f([a, b]) ⊇ [f(b), f(a)] .

Conclusion: 

When it comes to mathematics, a continuous function is defined as a real-valued function whose graph contains no breaks or holes. The concept of continuity serves as the conceptual underpinning for the intermediate value theorem and the extreme value theorem.