Continuidad

What is the continuity of a function at a point?

Continuity is a fundamental concept in mathematical analysis and plays a crucial role in the study of functions. There are three key conditions that determine whether a function is continuous at a specific point ( x = a ):

1. Definition of the function at the point: The function must be defined at ( x = a ). This implies that the value must exist within the domain of the function.

2. Existence of the limit: The limit of the function when ( x ) tends to ( a ) must exist. This requires that when applying algebraic operations to the limit, a given number is obtained.

3. Equality between limit and evaluation of the function: The evaluation of the function at the point must coincide with the value of the limit. Expressed mathematically, this is ( \lim_{x \to a} f(x) = f(a) ).

If any of these conditions is not satisfied, the function is not continuous at that point.

How to verify continuity in a practical example?

Let's see how to verify continuity with two specific examples:

Example 1: Quadratic function.

Consider the function ( f(x) = x^2 - 1 ) at the point ( x = 1 ).

1. Evaluation of the function:[ f(1) = 1^2 - 1 = 0 ] The function is well defined at ( x = 1 ).

2. Calculation of the limit:[ \lim_{x = 1} (x^2 - 1) = 1^2 - 1 = 0 ] The limit exists and is equal to 0.

3. Comparison of the limit and evaluation of the function: Both are equal to 0, so the function is continuous at ( x = 1 ).

Example 2: Rational function

Let's analyze the function ( f(x) = \frac{x+1}{x^2 - 1} ) at the point ( x = -1 ).

1. Evaluation of the function:[ f(-1) = \frac{-1 + 1}{(-1)^2 - 1} = \frac{0}{0} ] We have an indeterminacy ( \frac{0}{0} ), indicating an open point and therefore, lack of continuity.

This analysis reveals an infinitesimal jump at the point ( x = -1 ), resulting from an open point in the graph of the function. Functions that result in indeterminacies of the form ( \frac{0}{0} ) require further analysis to determine if continuity can exist through simplification techniques or l'Hôpital, although in this case I am verifying the function itself, not just the limit.

How to identify and solve discontinuities?

Identifying discontinuities is crucial when analyzing functions. Discontinuities appear when at least one of the three continuity conditions fails. In the example above, the rational function has an open point due to indeterminacy, indicating a discontinuity.

To strengthen your knowledge and skills in identifying discontinuities and continuity checks, it is essential:

• Practice solving functions through different computational methods.
• Understand the interpretation of the results and how they affect the behavior of the function.
• Participate in proposed exercises and share resolutions with classmates to improve understanding and learning.

I encourage you to look for support materials and solve challenges associated with the topic, sharing your experiences and solutions with your classmates to enrich the learning process.