Proving Continuity Using the Definition of Limits

Understanding the behavior of a function around a specific point is crucial in mathematical analysis. A function ?(x) is said to be continuous at a point x_0 if it satisfies the condition defined through the epsilon-delta definition of limits. This article delves into how the definition of limits can be harnessed to prove the continuity of a given function at a specific point. We will explore the conditions, the process, and the significance of these concepts in a comprehensive manner.

Introduction to Continuity and Limits

Continuity and limits form the bedrock of calculus. In simple terms, a function ?(x) is continuous at a point x_0 if three conditions are met:

?(x) is defined at x_0 limx→x_0 ?(x) exists limx→x_0 ?(x) ?(x_0)

These conditions boil down to the epsilon-delta definition of limits, which will be the cornerstone of our discussion.

The Epsilon-Delta Definition of Limits

The epsilon-delta definition of limits is a powerful analytical tool that formalizes the idea of a function approaching a certain value. Formally, limx→x_0 ?(x) L if, for every ε 0, there exists a δ 0 such that |?(x) - L| ε whenever 0 |x - x_0| δ.

Proving Continuity Using Limits

To prove that a function ?(x) is continuous at a point x_0, we need to show that the limit of ?(x) as x approaches x_0 equals the value of the function at x_0. This involves two main steps:

Identify the limit: Compute limx→x_0 ?(x). Apply the definition: Find a δ 0 for any given ε 0 such that |?(x) - ?(x_0)| ε whenever 0 |x - x_0| δ.

Step 1: Identifying the Limit

For our example, we need to determine whether the limit of some given function exists and what its value is as x approaches x_0. Suppose we are considering the function ?(x) x2 and want to prove it is continuous at x_0 2.

First, we compute the limit:

limx→2 x2 22 4

So, the limit of x2 as x approaches 2 is 4.

Step 2: Applying the Definition

Now we need to show that for every ε 0, there exists a δ 0 such that |x2 - 4| ε whenever 0 |x - 2| δ.

We can start by manipulations:

|x2 - 4| ε is equivalent to |(x-2)(x 2)| ε.

If we assume |x - 2| δ, then we can bound |x 2|. To do this, we can use the triangle inequality:

|x 2| ≤ |x - 2| |4| δ 4.

Thus, we can write:

|(x - 2)(x 2)| δ(δ 4).

We want δ(δ 4) ε. This can be achieved by selecting δ such that δ(δ 4) ε. For example, if we choose δ min(1, ε/5), we can ensure that the inequality holds because 1 δ and δ 4 5, making the product less than ε.

Conclusion

By using the epsilon-delta definition of limits, we have provided a rigorous proof that the function ?(x) x2 is continuous at x? 2. This method is not only applicable to polynomials but also to a wide variety of functions.

Understanding and applying the concept of limits to prove continuity is essential in mathematical analysis and forms the foundation of more advanced topics such as derivatives and integrals. Mastering this technique not only enhances your mathematical skills but also prepares you for deeper explorations in calculus and real analysis.

References

[1] Simmons, G. F. Introduction to Topology and Modern Analysis. McGraw-Hill, 1963.

[2] Courant, R. Differential and Integral Calculus I. Wiley, 1937.

[3] Apostol, T. M. Mathematical Analysis. Addison-Wesley, 1974.