Problem 19 - Calculus: Early Transcendentals

Question

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\sqrt{x-2} ; a=1$$

Step-by-Step Solution

Answer

Answer: No, the function is not continuous at the point $$a=1$$.

Step 1: Condition 1: Check if $$f(a)$$ exists

Plug the value of $$a=1$$ into the function $$f(x)$$ to see if $$f(a)$$ exists. $$f(1) = \sqrt{1 - 2} = \sqrt{-1}$$ Since the square root of a negative number is undefined in the real number system, $$f(a)$$ does not exist.

Step 2: Conclusion

Since the first condition of continuity is not met, there is no need to check the other conditions. The function $$f(x) = \sqrt{x-2}$$ is not continuous at the point $$a=1$$.

Key Concepts

Limits of Functions

In the realm of calculus, understanding the concept of limits is foundational to the study of continuity. A limit describes the value that a function f(x) approaches as the input x approaches a certain point. Formally, we express this as \( \lim_{x \to c} f(x) = L \), where \( L \) is the value the function is approaching as \( x \) gets closer to \( c \).

It's important to note that the function f(x) does not necessarily need to equal L when x equals c; it's the value f(x) is getting increasingly close to. This subtlety is crucial when dealing with functions that are not defined at some specific points but can still have a limit at those points. Understanding this helps in analyzing the behavior of functions and setting the stage for determining their continuity.

Continuity Checklist

To establish whether a function is continuous at a point, there's a straightforward continuity checklist to follow. For a function \( f(x) \) to be continuous at a point \( a \), three conditions must be satisfied:

• Existence: First, \( f(a) \) must exist. In other words, the function must be defined at \( a \).
• Limit: Second, the limit of \( f(x) \) as \( x \) approaches \( a \) must exist. This means that as \( x \) gets closer to \( a \) from both the left and the right, \( f(x) \) should approach the same value.
• Equality: Finally, the value of \( f(a) \) and the limit of \( f(x) \) as \( x \) approaches \( a \) must be equal. This ensures that there is no 'jump' or 'gap' in the function at that point.

Applying this checklist to the problem at hand, we immediately see that the function \( f(x) = \sqrt{x - 2} \) fails the first condition at \( a = 1 \), since \( f(1) \) does not exist within the real numbers. This single failing makes the function discontinuous at that point without needing to check the other two conditions.

Piecewise Functions

A piecewise function is a function that is defined by different expressions over different parts of its domain. Each 'piece' of the function has its own rule, and the full function can jump or transition smoothly between these rules at defined points. Analyzing the continuity of piecewise functions requires careful consideration of these transition points.

For each segment of the domain, the function can be treated individually, but special attention is needed at the 'borders' where the pieces meet. The continuity checklist should be applied at each of these border points to determine if the function is continuous there. In many cases, piecewise functions are constructed to be continuous, but it is still essential to verify continuity using the proper checklist for each specific case. Knowing how to work with piecewise functions is essential since they can model complex situations that are not adequately represented by a single, uniform rule.