Compare and contrast continuity of a function?

In summary, continuity for a function means that the graph of the function is a single, unbroken curve with no gaps or jumps. It is different from differentiability, which requires a well-defined derivative at every point. A function can be continuous at a specific point but not on an interval, and determining continuity at a point involves checking if the limit of the function exists and is equal to the value of the function at that point. Common types of discontinuities include removable, jump, and infinite discontinuities.
  • #1
javier007
2
0
PLEASE help me. I need to analyze the continuity of the piecewise function

f(x) = { sin(1/x) when x is not = to 0
_____{ 0_____ when x = 0

so i know sin(1/x) doent have a value at 0 but the second part of the function places the value of 0 at 0...BUT are both parts connected without any gaps, holes, or jumps in bewteen? in other words is it continuous or not?

I would appreciate it A LOT if you could explain it to me why or why not

thanks
 
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  • #2
Do you know how to tell if a function is continuous at a point?
 

1. What is the definition of continuity for a function?

Continuity for a function means that the graph of the function is a single, unbroken curve with no gaps or jumps. In other words, as the input values (x) approach a particular point, the output values (y) also approach a particular value.

2. How is continuity different from differentiability?

Continuity and differentiability are related concepts, but they are not the same. A function is continuous if its graph is a single, unbroken curve. On the other hand, a function is differentiable if it has a well-defined derivative at every point. In other words, a function can be continuous without being differentiable, but a function cannot be differentiable without being continuous.

3. Can a function be continuous at a point but not on an interval?

Yes, a function can be continuous at a specific point, but not on an interval. This means that the function is continuous at that point, but it has at least one point within the interval where it is not continuous. In this case, we say that the function has a point of discontinuity within the interval.

4. How do you determine if a function is continuous or discontinuous at a point?

A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, the left and right limits must be equal, and the function must be defined at that point. If the limit does not exist or is not equal to the value of the function, then the function is discontinuous at that point.

5. What are some common types of discontinuities in a function?

Some common types of discontinuities in a function include removable, jump, and infinite discontinuities. Removable discontinuities occur when there is a hole or gap in the graph that can be filled in to make the function continuous. Jump discontinuities occur when there is a sudden change in the function's value at a point. Infinite discontinuities occur when the limit of the function at a point is either positive or negative infinity.

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