Problem 1 Let $\phi$ be a monotone funct... [FREE SOLUTION]

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Measure Theory and Integration

G de Barra

$Math Studyset 91影视 Explanations$ Math

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Chapter 1: Problem 1

Let $\phi$ be a monotone function defined on $[a, b]$. Show that $\phi(a+)$ and $\phi(b-)$ exist.

Short Answer

Expert verified

Both limits $\phi(a+)$ and $\phi(b-)$ exist for monotone functions on $[a, b]$.

Step by step solution

Understanding Monotone Functions

A monotone function is one that is either entirely non-increasing or non-decreasing. This means, as the input values increase across the interval \[a, b\], the output values follow a consistent increasing or decreasing trend without any oscillation.

Left Limit at the Endpoint a

The left-hand limit $\phi(a+)$ is the limit of $\phi(x)$ as $x$ approaches $a$ from the right. Since $\phi$ is monotone on \[a, b\], it means $\phi(x)$ has a consistent pattern as $x$ moves closer to $a $ from inside the interval. Therefore, $\phi(a+)$ exists because the values $\phi(x)$ cannot oscillate.

Right Limit at the Endpoint b

The right-hand limit $\phi(b-)$ is the limit of $\phi(x)$ as $x$ approaches $b $ from the left. For a monotone function, the values approach this point consistently without interim fluctuations, ensuring that the limit exists. Thus, since $\phi$ is monotone from left towards $b$, $\phi(b-)$ also exists.

Conclusion

Since both $\phi(a+)$ and $\phi(b-)$ exist due to the monotonicity of $\phi$, it demonstrates the existence of limits at either endpoint of the interval for monotone functions

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit at Endpoint

A limit at an endpoint refers to the behavior of a function as it approaches a boundary of its interval.
For a function $ \phi(x) $ defined on $ [a, b] $, endpoints are the values $ a $ and $ b $.
There are specific terms to discuss limits at these points:

$ \phi(a+) $ describes the limit of $ \phi $ as $ x $ approaches $ a $ from the right.
$ \phi(b-) $ describes the limit as $ x $ approaches $ b $ from the left.

Monotone functions, whether increasing or decreasing, guarantee that these limits exist.
This is due to the function's consistent growth or reduction without any sudden jumps or oscillations.
Thus, as $ x $ approaches either $ a $ from the right or $ b $ from the left, the values of $ \phi(x) $ settle into a stable pattern, allowing these endpoint limits to be defined.

Non-decreasing Function

A non-decreasing function is one that never decreases as its input progresses.
As $ x $ moves along the interval $ [a, b] $, the output of the function either increases or remains constant.
This means:

For any two points $ x_1 $ and $ x_2 $ within the interval with $ x_1 < x_2 \, $, it holds that $ \phi(x_1) \leq \phi(x_2) $.
The absence of decreases ensures the function doesn鈥檛 have ups and downs, making it easier to predict changes in value as $ x $ progresses towards either endpoint.

This nature ensures that endpoints limits, $ \phi(a+) $ and $ \phi(b-) $, exist just like for any monotone function.
As there is no decline, the value is consistently heading in one direction or staying stable.

Non-increasing Function

A non-increasing function is characterized by its constant or decreasing output as its input value increases across a given interval.
This means that:

If you take any two points $ x_1 $ and $ x_2 \, $ where $ x_1 < x_2 \, $, then $ \phi(x_1) \geq \phi(x_2) $.
The function either goes downward or stays flat without any increases.

This ensures a stable behavior of the function, which directly impacts the limits at endpoints.
Just like a non-decreasing function, a non-increasing function will also create stable endpoint limits, such as $ \phi(a+) $ and $ \phi(b-) $.
This is because the function smoothly approaches these endpoint values without any jumps or reversals.

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91影视

Let \(\phi\) be a monotone function defined on \([a, b]\). Show that \(\phi(a+)\) and \(\phi(b-)\) exist.

Short Answer

Step by step solution

Understanding Monotone Functions

Left Limit at the Endpoint a

Right Limit at the Endpoint b

Conclusion

Key Concepts

Limit at Endpoint

Non-decreasing Function

Non-increasing Function

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