Question

(a) Prove: If \(f\) is continuous at \(x_{0}\) and \(f\left(x_{0}\right)>\mu,\) then \(f(x)>\mu\) for all \(x\) in some neighborhood of \(x_{0}\). (b) State a result analogous to (a) for the case where \(f\left(x_{0}\right)<\mu\). (c) Prove: If \(f(x) \leq \mu\) for all \(x\) in \(S\) and \(x_{0}\) is a limit point of \(S\) at which \(f\) is continuous, then \(f\left(x_{0}\right) \leq \mu\). (d) State results analogous to (a), (b), and (c) for the case where \(f\) is continuous from the right or left at \(x_{0}\).

Short answer

Question: Prove the following results for continuous functions and limit points: a) If $f$ is continuous at $x_0$ and $f(x_0) > \mu$, then there exists a neighborhood of $x_0$ such that $f(x) > \mu$ for all $x$ in that neighborhood. b) Analogous result if $f(x_0) < \mu$. c) If $f$ is continuous at a limit point $x_0$ of a set $S$, and $f(x) \leq \mu$ for all $x$ in $S$, then $f(x_0) \leq \mu.$ d) State analogous results for functions continuous from the right or left at a given point.

Step-by-step solution

Use the definition of continuity at a point

Since \(f\) is continuous at \(x_0\), we have \(\lim_{x\rightarrow x_{0}} f(x) = f(x_{0})\). Given \(f(x_0)>\mu\), we need to show that \(f(x)>\mu\) for all \(x\) in some neighborhood of \(x_{0}\).

Choose an arbitrary positive \(\epsilon\)

By definition of the limit, we can choose an arbitrary positive \(\epsilon\) such that for any \(0<|x - x_0|<\delta\), we have \(|f(x) - f(x_0)|<\epsilon\). We will later decide what \(\epsilon\) to pick.

Use the given inequality to choose \(\epsilon\) and find \(\delta\)

Let \(\epsilon = f(x_0) - \mu\). Then, \(|f(x) - f(x_0)|0\) such that \(0<|x - x_0|<\delta\).

Show \(f(x)>\mu\) for all \(x\) in the chosen neighborhood

Since \(|f(x) - f(x_0)|\mu\) for all \(x\) in the interval \((x_0-\delta, x_0+\delta)\). ##Part b: State an analogous result## The analogous result is: If \(f\) is continuous at \(x_{0}\) and \(f(x_{0})<\mu,\) then \(f(x)<\mu\) for all \(x\) in some neighborhood of \(x_{0}\). ##Part c: Prove the given statement for continuous functions and limit points##

Use the definition of continuity and limit point

Since \(f\) is continuous at the limit point \(x_0\), we have \(\lim_{x\rightarrow x_{0}} f(x) = f(x_{0})\). Given that \(f(x) \leq \mu\) for all \(x\) in \(S\), we need to show that \(f(x_{0}) \leq \mu\).

Assume that \(f(x_0)>\mu\)

Suppose, for contradiction, that \(f(x_0)>\mu\). Then, by part (a), there exists a neighborhood around \(x_0\) such that \(f(x)>\mu\) for all \(x\) in that neighborhood.

Show the contradiction and conclude

However, this contradicts the fact that \(f(x)\leq \mu\) for all \(x\) in \(S\), which includes points in the neighborhood of the limit point \(x_0\). Hence, our assumption must be false, and we conclude that \(f(x_{0}) \leq \mu\). ##Part d: State analogous results for right and left continuity## For right continuity: (a) If \(f\) is continuous from the right at \(x_0\) and \(f(x_0)>\mu,\) then \(f(x)>\mu\) for all \(x\) in some interval \((x_0, x_0+\delta)\). (b) If \(f\) is continuous from the right at \(x_0\) and \(f(x_0)<\mu,\) then \(f(x)<\mu\) for all \(x\) in some interval \((x_0, x_0+\delta)\). (c) If \(f(x) \leq \mu\) for all \(x\) in \(S\), and \(x_0\) is a limit point from the right of \(S\) at which \(f\) is continuous from the right, then \(f(x_{0}) \leq \mu\). For left continuity: (a) If \(f\) is continuous from the left at \(x_0\) and \(f(x_0)>\mu,\) then \(f(x)>\mu\) for all \(x\) in some interval \((x_0-\delta, x_0)\). (b) If \(f\) is continuous from the left at \(x_0\) and \(f(x_0)<\mu,\) then \(f(x)<\mu\) for all \(x\) in some interval \((x_0-\delta, x_0)\). (c) If \(f(x) \leq \mu\) for all \(x\) in \(S\), and \(x_0\) is a limit point from the left of \(S\) at which \(f\) is continuous from the left, then \(f(x_{0}) \leq \mu\).

Key concepts

Limit Point

In mathematics, a limit point (or accumulation point) of a set refers to a value that can be "approached" by values within the set. This means for any specified distance away from a limit point, however small, there is another point within the set distinct from the limit point itself. It’s like being on a street where you can never find a gap as each point hosts a home.
This concept is essential in understanding when a function's behavior is approaching a particular value without actually being that value in the set. Understanding limit points helps in following the behavior of functions that skirt around particular values as you move through the set.

Neighborhood

A neighborhood of a point in a topological space refers to a set containing, at a minimum, an open interval surrounding that point. You can picture it like a bubble around a point x in number line space, where all points within this bubble are considered to be in the set neighborhood of x.
Imagine standing at a specific point on a beach with your towel defining your space. Everything the towel touches is part of your beach 'neighborhood'. In mathematics, this ensures that any exploration of a point's immediate surroundings is sufficiently covered in a space.

• Neighborhoods help in understanding concepts like continuity and limits.
• They show us how changing a point slightly still adheres to the overall properties of a function.

Continuous Functions

Continuous functions are functions without breaks, jumps, or holes at any point in their domain. These functions maintain a steady flow of values with no surprises. A common metaphor is comparing it to driving smoothly on a highway without any sudden stops or detours.
To be continuous at a point x = x_0, a function f must satisfy the condition: \[\lim_{{x\to{x_0}}} f(x) = f(x_0)\]
This means that as x approaches x_0, f(x) approaches the value of f at x_0. No sudden shifts occur, making it predictable and thus useful in mathematical analysis.

Epsilon-Delta Definition

The epsilon-delta definition is a mathematical way to say that a function f is continuous at a point x = x_0. It states:\[\forall \epsilon > 0, \exists \delta > 0 \text{ such that if } 0 < |x - x_0| < \delta, \text{ then } |f(x) - f(x_0)| < \epsilon.\]
Breaking this down: for every small window of error (epsilon), there exists a range (delta) around x_0, where all points inside this range yield function values close to { f(x_0)}.
Think of it like saying how precise you want to be when measuring something and finding out how consistent you can be within a given area. Epsilon-delta gives us a precise mechanism to define, verify, and work with continuity in a mathematical framework.