**Introduction to Continuity in Real Analysis**

In the realm of real analysis, the concept of continuity is fundamental. Continuity refers to the smooth flow of a graph without breaks or jumps. In simpler terms, a function is continuous at a given point if small changes in the input result in small changes in the output. This article delves into the techniques and methods for proving continuity at a point, with a focus on rigorous mathematical proofs.

Understanding Continuity

Before we dive into the methods of proving continuity, it is important to understand what it means for a function to be continuous. A function ( f(x) ) is continuous at a point ( x c ) if the following three conditions are met:

Existence of the Limit: The limit of the function as ( x ) approaches ( c ) exists. In mathematical notation, this is written as ( lim_{x to c} f(x) ) exists. Function Value: The value of the function at the point ( c ) exists and is equal to the limit. In mathematical notation, this is ( f(c) ) exists and ( f(c) lim_{x to c} f(x) ). Equality: The limit from the left and the limit from the right are equal at point ( c ).

These conditions together ensure that the function does not have any breaks or jumps at the point ( c ).

Proving Continuity at a Point

To prove that a function ( f(x) ) is continuous at a point ( x c ), we need to verify the three conditions mentioned above. Let's illustrate this with a simple example.

Example Function

Consider the function ( f(x) |x| ). We want to prove that ( f(x) ) is continuous at ( x 0 ).

Step-by-Step Proof

Step 1: Existence of the Limit

We need to show that ( lim_{x to 0} f(x) ) exists. The function ( f(x) |x| ) means that ( f(x) x ) for ( x geq 0 ) and ( f(x) -x ) for ( x [ lim_{x to 0} f(x) 0 ]

Step 2: Function Value

We need to check if ( f(0) ) is defined and equals the limit we calculated in Step 1. For the function ( f(x) |x| ),

[ f(0) 0 ]
Since ( f(0) 0 ) and ( lim_{x to 0} f(x) 0 ), the second condition is satisfied.

Step 3: Equality of Limits

We need to verify that the limit from the right and the limit from the left are equal at ( x 0 ).

Left-hand Limit: As ( x ) approaches 0 from the left (i.e., ( x to 0^- )), [ lim_{x to 0^-} f(x) lim_{x to 0^-} (-x) 0 ] Right-hand Limit: As ( x ) approaches 0 from the right (i.e., ( x to 0^ )), [ lim_{x to 0^ } f(x) lim_{x to 0^ } x 0 ]

Since both the left-hand limit and the right-hand limit are equal to 0, the third condition is also satisfied.

Therefore, the function ( f(x) |x| ) is continuous at ( x 0 ).

Common Techniques for Proving Continuity

Using Limits

The method used in the example above involves directly evaluating the limits and comparing them to the value of the function at the point. This is a straightforward approach that works well when the function is defined piecewise or involves absolute value functions.

Using Epsilon-Delta Definition

A more rigorous method is the u03B5-u03B4 definition of continuity. According to this definition, a function ( f(x) ) is continuous at ( x c ) if for every (epsilon > 0), there exists a (delta > 0) such that for all ( x ) satisfying ( 0

The (epsilon-delta) definition allows us to prove continuity in a more general sense, particularly for functions that are not defined piecewise or involve more complex expressions. Here's a brief outline of how to prove continuity using the (epsilon-delta) definition:

Assume (epsilon > 0) is given. Find a (delta > 0) such that for all ( x ) satisfying ( 0 Verify that for any ( x ) satisfying ( 0

Using Limits at Infinity

If a function ( f(x) ) is defined for all ( x geq a ) for some ( a in mathbb{R} ), we can often prove its continuity by showing that the limit of ( f(x) ) as ( x ) approaches infinity exists. This is particularly useful for polynomial, rational, and trigonometric functions that are defined over a large domain.

Conclusion

Proving the continuity of a function is a crucial step in many mathematical analyses. By understanding the conditions for continuity and applying the appropriate techniques, you can ensure that your function is continuous at a given point. Whether you use limits or the (epsilon-delta) definition, the key is to be precise and rigorous in your approach. Understanding these methods will not only help you solve problems but also deepen your understanding of real analysis.