Continuous periodic piecewise differentiable

MHB
Thread starter Dustinsfl
Start date Oct 14, 2012
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Continuous Differentiable Periodic

In summary, continuous refers to the smoothness of a function's graph without abrupt changes, while periodic means the function repeats itself at regular intervals. A piecewise differentiable function is composed of smaller functions with different rules for specific intervals, while a regular differentiable function has a single rule for the entire domain. An example of a continuous periodic piecewise differentiable function is the sawtooth wave. These functions are useful in science for modeling real-world phenomena and allowing for more accurate calculations and predictions.

Oct 14, 2012

Dustinsfl

Suppose that $f(\theta)$ is a continuous periodic piecewise differentiable function. Prove that $f(\theta) = f(0) + \int_0^{\theta}g(t)dt$ for a piecewise continuous $g$.

I just need a nudge in the right direction here.

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Oct 20, 2012

tarantella

Can you do it when $f$ is differentiable? THen use the fact that there are only finite many points where $f$ is not differentiable.

Related to Continuous periodic piecewise differentiable

1. What does "continuous" mean in relation to continuous periodic piecewise differentiable functions?

Continuous refers to the property of a function where there are no abrupt changes or discontinuities in its graph. This means that the function can be drawn without lifting the pen from the paper.

2. What is the significance of "periodic" in a continuous periodic piecewise differentiable function?

Periodic means that the function repeats itself after a certain interval, called the period. This means that the function will have the same values at regular intervals along the x-axis, creating a repeating pattern.

3. How is a piecewise differentiable function different from a regular differentiable function?

A piecewise differentiable function is made up of smaller, simpler functions that are defined over specific intervals. This means that the function may have different rules or equations for different parts of the domain. A regular differentiable function, on the other hand, has a single rule or equation that applies to the entire domain.

4. Can you provide an example of a continuous periodic piecewise differentiable function?

One example is the sawtooth wave function, which is a continuous function that repeats itself in a "sawtooth" pattern. It is piecewise differentiable because it is made up of linear segments that are connected at specific points.

5. Why are continuous periodic piecewise differentiable functions useful in science?

Continuous periodic piecewise differentiable functions are useful in science because they can model real-world phenomena that exhibit repeating patterns, such as waves, vibrations, and oscillations. They also allow for more precise and accurate calculations and predictions compared to simpler functions.