Question

Continuity of the absolute value function Prove that the absolute value function $|x|$ is continuous for all values of $x.$ (Hint: Using the definition of the absolute value function, compute $\underset{x\to 0^{-}}{lim}|x|\text{ and }\underset{x\to 0^{+}}{lim}|x|.)$

Short answer

Question: Prove the continuity of the absolute value function $|x|$ for all values of $x$. Answer: The continuity of the absolute value function $|x|$ has been proven by calculating the left-hand limit and the right-hand limit as $x$ approaches 0. Since both limits are equal and equal to 0, the function is continuous at $x=0$. Given the even function property ($|x|=|-x|$), the function's behavior is symmetric around the y-axis. Hence, if the function is continuous at $x=0$, it is continuous for all values of $x$. Therefore, the absolute value function $|x|$ is continuous for all values of $x$.

Step-by-step solution

Calculate the left-hand limit of the absolute value function.

To find the left-hand limit as $x$ approaches 0, we write $\underset{x\to 0^{-}}{lim}|x|$. For $x<0$, $|x|=-x$. Hence, the limit will be $\underset{x\to 0^{-}}{lim}(-x)$.

Solve the left-hand limit.

Now, to solve the left-hand limit, we substitute x with values that are slightly smaller than 0 (e.g., -0.01, -0.001, etc.). As x approaches 0 from the left, the limit becomes $\underset{x\to 0^{-}}{lim}(-x)=0$.

Calculate the right-hand limit of the absolute value function.

To find the right-hand limit as $x$ approaches 0, we write $\underset{x\to 0^{+}}{lim}|x|$. For $x\ge 0$, $|x|=x$. Hence, the limit will be $\underset{x\to 0^{+}}{lim}(x)$.

Solve the right-hand limit.

Now, to solve the right-hand limit, we substitute x with values that are slightly larger than 0 (e.g., 0.01, 0.001, etc.). As x approaches 0 from the right, the limit becomes $\underset{x\to 0^{+}}{lim}(x)=0$.

Compare the left-hand limit and the right-hand limit.

We found that the left-hand limit is equal to the right-hand limit, as both are 0. Since the two limits are equal, the absolute value function is continuous at $x=0$.

Prove the continuity of the absolute value function for all x values.

We have proved the continuity of the absolute value function at $x=0$. Given that $|x|$ is an even function ($|x|=|-x|$), its behavior is symmetric around the y-axis; therefore, if the function is continuous at $x=0$, it will be continuous for all values of $x$. Thus, we can conclude that the absolute value function $|x|$ is continuous for all values of $x$.

Key concepts

Limits and Continuity

Understanding the concepts of limits and continuity is fundamental in calculus. A limit refers to the value that a function approaches as the input (or 'x' value) approaches a certain point. Continuity, on the other hand, implies that a function is smooth and unbroken over an interval; there are no jumps, holes, or vertical asymptotes.

When discussing continuity at a specific point, we focus on the behavior of the function as it gets close to that point from both sides. If the limits from the left and the right exist and are equal and the function actually equals that value at the point in question, the function is continuous at that point. This idea is essential in developing a rigorous foundation for calculus, as many of the theorems rely on functions being continuous.

Left-hand Limit

The left-hand limit of a function as x approaches a point 'a' from the left (denoted as 'a-') is the value that the function approaches as we get arbitrarily close to 'a' from values less than 'a'. It's like sneaking up on the point from the negative side on a number line.

To calculate this, we look at values of x that are just slightly less than 'a' and see what y-values our function is approaching. If these y-values approach a single number, then that is the left-hand limit. This concept is not just an abstract idea—it's used to understand behavior such as reaction rates in chemistry or the moment before a material undergoes stress in physics, among others.

Right-hand Limit

In contrast, the right-hand limit of a function as x approaches a point 'a' from the right (denoted as 'a+') is the value the function is approaching as x approaches 'a' from values greater than 'a'. Imagine approaching the point on a graph from the positive side.

If the y-values of the function get closer and closer to a certain number as we substitute x-values increasingly near to 'a' from the right, that number is the right-hand limit. Grasping this concept allows us to predict behaviors like the decay of a substance in biology, or the diminishing returns in economics as resources become scarce.

Continuity Proof

The continuity proof is a logical argument that establishes whether a function is continuous at a point, over an interval, or everywhere. To show that a function like the absolute value function is continuous, we must demonstrate that there are no discontinuities—no jumps, breaks, or holes.

This typically involves checking three conditions:

• The function is defined at the point of interest.
• The limit of the function as x approaches the point exists.
• The function's value at that point equals the limit.

When all three are satisfied, the function is continuous at that point. Proofs like this are the underpinnings of calculus, ensuring that the functions studied behave predictably and that the rules of calculus can be applied to them.