Algebra of Continuous Function

The algebra of continuous functions uses four arithmetic operations: addition of continuous functions, subtraction of continuous functions, multiplication of continuous functions, and division of continuous functions. In the algebra of continuous functions, continuous functions are used in equations involving binary operations of any kind.

Assume that f(x) and g(x) are continuous functions at x = a. Accordingly, we find the following rules:

• When x = 0, f + g is continuous,
• When x = x0, f – g is continuous,
• When x = x0, f . g is continuous and,
• A continuous function f/g exists at x = x0 (g(x) ≠ 0).
• Continuity theorem based on composite functions

In the case where f is continuous at g (x0) and g is continuous at x0, then fog is continuous at x0.

Addition and Subtraction of Two Continuous Functions

• The addition of a continuous function, At x = x0, f + g is continuous.
• Subtraction of Continuous Function, at x = x0, f – g is continuous

We need to determine whether (f(x) + g(x)) is continuous at x = a. 

Thus, we need to make sure that the three conditions of continuity are met. Due to the fact that f (x) and g (x) are continuous at x = a, the following three conditions are satisfied.

f(a) and g(a) are defined

lim x→a f(x) = f(a) = k1 (say) and 

limx→ag(x) = g(a) = k2 (say)

Using them, we will get:

=> [f(a) + g(a)] is clearly defined at x = a because both f(a) and g(a) are defined.

=> Using the Summation Law of limits i.e. The limit of a sum is that the sum of the limits; 

we will get:

limx→a [f(x) + g(x)] = lim x→a f(x) + limx→a g(x) = k1 + k2 (here)

=> f(a) + g(a) = k1 + k2 = limx→a [f(x) + g(x)]

This implies that the function [f (x) + g(x)] is continuous at x = a. To prove the subtraction rule, you would replace the + sign with a – sign, as in the proof for the addition rule.

Multiplication and Division of Two Continuous Functions

• The Multiplication of Continuous Functions, f . g is continuous at x = x0,

• a division of continuous functions, At x = x0, f/g is continuous (g(x) ≠ 0)

Using the merchandise Law of limits i.e. The limit of a product is that the product of the limits; we will get:

limx→a [f(x) × g(x)] = lim x→a f(x) × limx→a g(x) = k1 × k2 (here)

Using the Quotient Law of limits i.e. The limit of a quotient is that the quotient of the limits; we will get:

limx→a [f(x)/g(x)] = lim x→a f(x)/limx→a g(x) = k1/k2 (here, provided k2 ≠ 0)

The proofs will be presented in a similar manner.

Continuous Function

The concept of a continuous function is a relationship such that a continuously changing argument induces a continuously changing value. In other words, there are no abrupt changes in value that are referred to as discontinuities. Functions are continuous when they do not experience abrupt changes in values. A continuous function has no restrictions over the course of its domain or interval. The continuous nature of a function at a given point can be established by its definition at that point, by its limit existing at that point, and by its value at that point equaling the value of the limit.

Discontinuous Function

In a domain where a function is not continuous at a certain point, it is said to have a discontinuity at that point. Depending on the nature of the function, this set could be discrete, dense, or it could even encompass the entire domain. Functions that are discontinuous are those in which there is a gap or jump between graph points. It represents an area in which the graph cannot be continued without moving somewhere else. A discontinuous function f (x) is defined as one that is not continuous at x = a.

Conclusion

In the study of optimization problems, continuous functions play a very important role. In machine learning algorithms and optimization methods, continuous functions have many practical applications because of their properties. According to the extreme value theorem, a function has a maximum value at some point within a given interval. Similarly, a function has a minimum value. The concept of continuous functions is of the utmost importance in mathematics and functions. Nevertheless, not all functions are continuous, and those are the discontinuous ones.