Continuity of |f|: Examples w/ Discontinuity at 0

Thread starter sean/mac
Start date Apr 30, 2008
Tags

Continuity

In summary, continuity of a function is the smoothness or connectedness of its graph, with no breaks or jumps. It is important for understanding a function's behavior and using calculus to find its derivative and integral. Some examples of functions with discontinuity at 0 include 1/x, |x|, and a piecewise function. To determine if a function is continuous at a specific point, the limit must exist and be equal to the function's value at that point. A function can have multiple points of discontinuity, such as f(x) = x^2 at x = 0 and x = 2.

Apr 30, 2008

sean/mac

Give an example of a function f which is discontinuous at 0 yet abs(f ) is continuous at 0

i have tried for an hour or so trying to think of one, even hints would be helpful

Physics news on Phys.org

Apr 30, 2008

nicksauce

Science Advisor

Homework Helper

I can think of a very trivial example... sort of like a step function.

FAQ: Continuity of |f|: Examples w/ Discontinuity at 0

What is continuity of a function?

Continuity of a function refers to the smoothness or connectedness of a function's graph. A function is considered continuous if its graph has no breaks or jumps.

What is the importance of continuity of a function?

Continuity is important for understanding the behavior of a function and making predictions about its values. It also allows us to use calculus to find the derivative and integral of a function.

What are some examples of functions with discontinuity at 0?

Some examples of functions with discontinuity at 0 include:

f(x) = 1/x
f(x) = |x|
f(x) = 1 if x is rational, 0 if x is irrational

How do you determine if a function is continuous at a specific point?

A function is continuous at a specific point if the limit of the function as x approaches that point exists and is equal to the function's value at that point. In other words, the left and right limits must be equal and equal to the function's value at that point.

Can a function have multiple points of discontinuity?

Yes, a function can have multiple points of discontinuity. For example, the function f(x) = x^2 has a point of discontinuity at x = 0 and x = 2. It is still considered continuous on its domain, except at these two points.