Algebra of Continuous Functions

This article aims to provide students with easy-to-understand and well-written study material notes on the topic – Algebra of Continuous Functions. This study guide will help students grasp the concepts easily and prepare for CBSE Classes 11th & 12th examinations. 

The continuity of a function is determined by the limits of the function of interest, so it is reasonable to expect similar results to the limits. It also mentions configuration rules that you may not be aware of but are very important for future use. 

Overview of Algebra of Continuous Functions

The following is an overview of the Algebra of Continuous Functions.

Suppose f (x) and g (x) are two continuous functions at point x = a. Then there are the following rules: 

Addition and subtraction rules in Algebra of Continuous Functions

Prove that f (x) + g (x) is continuous at x = a and f (x) – g (x) is continuous with x = a.

Proof: We need to check the continuity of (f (x) + g (x)) with x = a. 

Therefore, we need to ensure that the three continuity conditions are met.

Note that the functions f (x) and g (x) are continuous at x = a. Hence, the three continuity conditions are automatically met.

 f (a) and g (a) are defined as Lim x → a f (x) = f (a) = k1 (for example) and Lim x → a g (x) = g (a) = k2 (for example).

Derivation of parametric function Derivative of implicit function Derivative of inverse trigonometric function Exponential and logarithmic functions. Logarithmic Derivative Mean value theorem Second Derivative with these, you get: 

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

-> Use the limit sum law d. NS. The total limit is the sum of the limits. It will be as follows. 

 Limx → a [f (x) + g (x)] = Limx → af (x) + Limx → ag (x) = k1 + k2 (here) 

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

 Therefore, the function [f (x) + g (x)] is continuous with x = a. The proof of the subtraction rule is similar to the proof of the addition rule (just replace the + sign with the-sign). 

Multiplication and division rules  in Algebra of Continuous Functions

f (x) × g (x) is continuous with x = a f (x) / g (x) is continuous with x = a. Assuming g (a) ≠ 0 Proof: Applying the rule of the product of limits, d. NS. The product limit is the product of the limits. 

 Lim x → a [f (x) × g (x)] = Lim x → a f (x) × Lim x → a g (x) = k1 × k2 (here) Use the quotient law of the limit value is a marginal quotient. 

 Lim x → a [f (x) / g (x)] = Lim x → a f (x) / Lim x → a g (x) = k1 / k2 (here, k2 ≠ 0 is assumed) 

 Next, the proof continues. There is. Next, look at the resolved question (1), which shows the applicability of one of these rules. 

configuration rule f (g (x)) and g (f (x)) are continuous at x = a.

Continuous function definition

For a function to be continuous at a point, it must be defined at that point, its limit exists at that point, and the value of the function at that point. This is called the continuous function definition.

Continuous definition math

A continuous function is a function that has no limits over the entire range or at specific intervals. The graphs do not contain any signs of asymptote or discontinuity. 

Continuous function examples

On the other hand, if one has to consider continuous function examples, a continuous function is a function that can take any number within a certain interval. For example, if a continuous function is 1 at one point and 2 at another, the continuous function is 1.5 at another point. Continuous functions always connect all their values, but discrete functions have separation.

Conclusion

Algebra of Continuous Functions additionally recognises approximately a composition rule that may not be acquainted with you. However, it could be very vital for destiny applications. The continuity of a function is determined completely by the limits of the function of interest, so it is reasonable to expect similar results to the limits. It also mentions configuration rules that you may not be aware of but are very important for future use.