# Proving continuity of inverse cube function

• Eclair_de_XII
In summary: By setting ##|x-a|<1##, we can guarantee that ##|x|<1+|a|##, which gives us the necessary condition for the lemma. Then, using the definition of ##g(x)##, we can simplify the expression and get a bound for ##|g(x)|##. This allows us to choose a suitable ##\delta## in the proof. In summary, the proof given in two steps shows how to prove a lemma and then use it to prove a result. The first step involves defining a function and showing that it is bounded from below by a positive number. The second step uses the lemma to prove a result by setting a suitable value for ##\delta## in the proof.
Eclair_de_XII
Homework Statement
Let ##f:\mathbb{R}\longrightarrow \mathbb{R}## be defined by ##f(x)=\sqrt[3]{x}##. Prove that for all ##\epsilon>0##, there is a ##\delta>0## such that for ##a\in \mathbb{R}##, if ##|x-a|<\delta## for ##x\in \mathbb{R}##, then ##|\sqrt[3]{x}-\sqrt[3]{a}|<\epsilon##.
Relevant Equations
Difference of cubes formula: ##x^3-a^3=(x-a)(x^2+xa+a^2)##
No theorems allowed; must show by basic epsilon-delta proof.
The proof is given in two steps
1. Prove the lemma.
2. Use lemma to prove result.

%%1-Lemma%%
Assume ##a\neq0##. Define ##g:(-(|a|+1),|a|+1)\longrightarrow \mathbb{R}## by ##g(x)=\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}##. Then ##g## is bounded from below by some positive number ##m##.
---proof---
\begin{align}
g(x)&=&\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}\\
&=&\sqrt[3]{x^2}+\sqrt[3]{xa}+\frac{1}{4}\sqrt[3]{a^2}+\frac{3}{4}\sqrt[3]{a^2}\\
&=&\left(\sqrt[3]{x}+\frac{1}{2}\sqrt[3]{a}\right)^2+\frac{3}{4}\sqrt[3]{a^2}\\
&\geq&\frac{3}{4}\sqrt[3]{a^2}\\
&=&m
\end{align}

%%2-Attempt at Proof%%
Let ##\epsilon>0## and set ##\delta\leq\min\{1,\frac{\epsilon}{m}\}##. For ##a\neq0##, if ##|x-a|=|\sqrt[3]x-\sqrt[3]a|\left|\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}\right|<\delta##, then:

\begin{align}
1&>&|x-a|\\
&\geq&||x|-|a||\\
&\geq&|x|-|a|
\end{align}

This implies that ##|x|<1+|a|##, which gives the necessary condition for the lemma.

\begin{align}
\left|\sqrt[3]x-\sqrt[3]a\right|&=&\frac{|x-a|}{\left|\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}\right|}\\
&=&\frac{|x-a|}{\left|g(x)\right|}\\
&\leq&\frac{|x-a|}{m}\\
&<&\delta\cdot m\\
&\leq&\frac{\epsilon}{m}\cdot m\\
&<&\epsilon
\end{align}

If ##a=0##, then set ##\delta=\epsilon^3## so that if ##|x|<\delta##, then ##\left|\sqrt[3]x\right|<\delta^{\frac{1}{3}}=(\epsilon^3)^\frac{1}{3}=\epsilon##.

Last edited:
Couldn't we simply take ##|x-a| \lt 1## and then get a bound for ##|x^2 + xa + a^2|##?

Hall said:
Couldn't we simply take ##|x-a| \lt 1## and then get a bound for ##|x^2 + xa + a^2|##?
This is the usual approach for this and similar proofs.

## What is the inverse cube function?

The inverse cube function is a mathematical function where the input value is raised to the power of -3. In other words, it is the reciprocal of the cube of the input value. The inverse cube function can be represented as f(x) = 1/x^3.

## Why is proving continuity of the inverse cube function important?

Proving continuity of a function is important because it ensures that the function is well-behaved and has no sudden jumps or breaks. In the case of the inverse cube function, proving continuity is important because it allows us to use the function in various mathematical calculations and applications with confidence.

## What is the process for proving continuity of the inverse cube function?

The process for proving continuity of the inverse cube function involves showing that the function is continuous at every point in its domain. This can be done by showing that the limit of the function at a point is equal to the value of the function at that point, and by showing that the function is continuous from both the left and right sides of the point.

## What are the challenges in proving continuity of the inverse cube function?

One of the main challenges in proving continuity of the inverse cube function is dealing with points where the function is undefined, such as when the input value is 0. In these cases, special techniques, such as using the squeeze theorem, may be necessary to show continuity. Another challenge is the complexity of the function, which may require advanced mathematical knowledge to prove continuity.

## What are some real-life applications of the inverse cube function?

The inverse cube function has many real-life applications, including in physics, engineering, and economics. For example, it is used in calculating the force of gravity between two objects, the strength of electric fields, and the relationship between distance and volume in economics. It is also used in modeling natural phenomena, such as population growth and the spread of diseases.

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