How can I prove the continuity of $f$ at $x = 1$?

In summary, a $\varepsilon-\delta$ proof was given to show that the function $f(x) = x^2 + 3x - 3$ is continuous at $x = 1$. The proof involved setting $\delta$ to be the minimum of $1$ and $\frac{\varepsilon}{6}$ and showing that for any $\varepsilon > 0$, there exists a $\delta > 0$ such that $|x - 1| < \delta$ implies $|f(x) - f(1)| < \varepsilon$. This was achieved by manipulating the expression $|f(x) - f(1)| = |x^2 + 3x -
  • #1
Dustinsfl
2,281
5
Give a $\varepsilon-\delta$ proof that the function $f$ given by the formula $f(x) = x^2 + 3x - 3$ is continuous at $x = 1$.Given $\varepsilon > 0$.
There exist a $\delta > 0$ such that $|x - c| < \delta$ whenever $|f(x) - f(c)| < \varepsilon$.
From the statement of the $\varepsilon-\delta$ definition, we have that
$$
|x - 1| < \delta\quad\text{and}\quad |f(x) - f(1)| < \varepsilon.
$$
Let's look at $|f(x) - f(1)| = |x^2 + 3x - 4| < \varepsilon$.
Then
\begin{alignat*}{3}
|x^2 + 3x - 4| & = & |(x - 1)(x + 4)|\\
& = & (x + 4)|x - 1|
\end{alignat*}

What can I do about (x + 4)?
 
Physics news on Phys.org
  • #2
dwsmith said:
Give a $\varepsilon-\delta$ proof that the function $f$ given by the formula $f(x) = x^2 + 3x - 3$ is continuous at $x = 1$.Given $\varepsilon > 0$.
There exist a $\delta > 0$ such that $|x - c| < \delta$ whenever $|f(x) - f(c)| < \varepsilon$.
From the statement of the $\varepsilon-\delta$ definition, we have that
$$
|x - 1| < \delta\quad\text{and}\quad |f(x) - f(1)| < \varepsilon.
$$
Let's look at $|f(x) - f(1)| = |x^2 + 3x - 4| < \varepsilon$.
Then
\begin{alignat*}{3}
|x^2 + 3x - 4| & = & |(x - 1)(x + 4)|\\
& = & (x + 4)|x - 1|
\end{alignat*}
What can I do about (x + 4)?
Start with $0<\delta <1$
[tex]\begin{align*}|x-1|&< 1\\ 0<x &< 2 \\4< x+4 &<6\\|x+4|&< 6\end{align*}[/tex].
 
  • #3
Plato said:
Start with $0<\delta <1$
[tex]\begin{align*}|x-1|&< 1\\ 0<x &< 2 \\4< x+4 &<6\\|x+4|&< 6\end{align*}[/tex].

Do I have to consider for $\delta > 1$ next too?
 
  • #4
dwsmith said:
Do I have to consider for $\delta > 1$ next too?
NO! You let $\delta = \min \left\{ {1,\frac{\varepsilon }{6}} \right\}$
 

What is the definition of continuity of a function?

The continuity of a function refers to the property of a function where there are no sudden jumps or breaks in its graph. In other words, a function is continuous if it can be drawn without lifting the pencil from the paper.

What are the three main criteria for a function to be continuous?

A function must satisfy the following criteria to be considered continuous:
1. The function must be defined at that point.
2. The limit of the function at that point must exist.
3. The limit of the function at that point must be equal to the value of the function at that point.

How is continuity different from differentiability?

Continuity and differentiability are related but different concepts. A function is continuous if it can be drawn without lifting the pencil from the paper, while a function is differentiable if it has a derivative at every point. A function can be continuous but not differentiable, but it cannot be differentiable without being continuous.

What is the importance of continuity in calculus?

Continuity is an essential concept in calculus as it allows us to define the derivative and integral of a function. These two operations are fundamental in calculus and are used to solve a wide range of problems in mathematics, physics, and engineering.

How can we determine if a function is continuous using algebra?

To determine if a function is continuous using algebra, we need to check if the function satisfies the three criteria for continuity:
1. The function is defined at a given point.
2. The limit of the function at that point exists.
3. The limit of the function at that point is equal to the value of the function at that point.
If all three criteria are satisfied, then the function is continuous at that point. If any of the criteria are not met, then the function is not continuous at that point.

Similar threads

Replies
11
Views
1K
Replies
1
Views
758
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Calculus
Replies
25
Views
905
Replies
1
Views
830
  • Differential Equations
Replies
1
Views
717
Replies
5
Views
270
Replies
9
Views
891
  • Differential Equations
Replies
1
Views
624
Back
Top