Continuous Functions

What Is A Continuous Function?

Continuous functions are functions whose graphical representation doesn’t have any break or cuts along the curve or body of the graph. A continuous function can be understood simply by its name. The function is not broken and continuous. One basic requirement for any function to be a continuous function is that there should not be any discontinuities in between the domain values. 

To check whether a function is continuous or not? There will not be any sudden or abrupt break or cut upwards or downwards in the graph of a continuous function. If any function doesn’t satisfy these conditions entirely, then that function will be called a discontinuous function.

A function is continuous if it is possible for anyone to draw the graphical representation of that function in one go that means without lifting the pen.

A function f(x) is continuous at a point x = a if

1. f(a) should exist really
2. limₓ → ₐ f(x) must also exist

The “lim x tends to a f(x) exists” indicates that the function could perhaps approach the same value from either the left and right sides of the value x = a, and “lim x tends to a f(x) = f(a)” indicates that the function’s limit at x = an is the same as f(a). The function will be continuous at that time if these two conditions are met.

Properties of a Continuous Function

Here are a few properties of function continuity. At x = u, if two functions A(x) and B(x) are continuous, then

At x = u, A + B, A – B, and AB are all continuous.

At x = u, A/B is similarly continuous if B(u) = 0.

The compositional function (A o B) is also consistent at x = un if A is continuous at B(u).

Over the collection of all real numbers, all polynomial functions are continuous.

Over the collection of all real numbers, the absolute value function |x| is continuous.

At all real figures, exponential functions are continuous.

At all real figures, the trigonometric functions like sin u and cos u are continuous.

Other trigonometric functions on their respective domains, such as tan u, cosec u, sec u, and cot u are continuous.

Continuous Function Theorems

Continuous functions have a major role to play in defining the applications of the derivative section. Continuous functions have several theorems to reduce the time complexity of the problem. Here are some fundamental important theorems.

Theorem 1: The polynomial functions in the form of ax + b are continuous in the range from minus infinity to positive infinity (-∞, ∞).

Theorem 2: The exponential function such as ex, and trigonometric functions such as sin x, tan x, cos x,  are continuous in the range from minus infinity to positive infinity on (-∞, ∞).

Theorem 3: If two functions A and B are continuous on an interval [p, q], then A+B, A-B, and AB are continuous functions within the interval [p, q]. But A/B is continuous on [p, q] given that A/B is not equal to zero anywhere in the [p, q] interval.

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. As per the rules stated in the extreme value theorem, this 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.