Absolute Values and Continuous Functions

In summary, the conversation discusses the continuity of a function and its absolute value. It is mentioned that if a function is continuous, its absolute value is also continuous, but the reverse is not always true. An example is given of a function that is discontinuous but its absolute value is continuous.
  • #1
Rosey24
12
0

Homework Statement



We recently proved that if a function, f, is continuous, it's absolute value |f| is also continuous. I know, intuitively, that the reverse is not true, but I'm unable to come up with an example showing that, |f| is continuous, b f is not. Any examples or suggestions would be appreciated. Thanks!

Homework Equations



The Attempt at a Solution

 
Physics news on Phys.org
  • #2
What about [itex]f(x) = -1[/itex] if [itex]x < 0[/itex] and 1 if [itex]x \ge 0[/itex]?
Then [itex]|f(x)| = 1[/itex] for all x - about as continuous as you get them, but f is not. In fact, you could even do something pathetic like make f equal -1 on all rationals and 1 on all irrational numbers :)
 

1. What is an absolute value?

An absolute value is the distance of a number from zero on a number line. It is always a positive number, regardless of the sign of the original number.

2. How do you find the absolute value of a number?

To find the absolute value of a number, you can simply remove the negative sign if it exists. Alternatively, you can use the absolute value function in most scientific calculators, which is denoted by "|" around the number.

3. What is a continuous function?

A continuous function is a function whose graph is an unbroken curve with no gaps or holes. This means that there are no sudden jumps or breaks in the graph, and every point on the graph is connected.

4. How do you determine if a function is continuous?

A function is continuous if it satisfies three conditions: 1) the function is defined at the point, 2) the limit of the function at that point exists, and 3) the limit of the function at that point is equal to the value of the function at that point. If all three conditions are met, the function is continuous.

5. Can a continuous function have a vertical asymptote?

No, a continuous function cannot have a vertical asymptote. A vertical asymptote occurs when the function approaches infinity or negative infinity at a certain point, which would create a break in the graph and violate the condition of continuity.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
324
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
911
  • Calculus and Beyond Homework Help
Replies
4
Views
308
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
Back
Top