MHB Continuous periodic piecewise differentiable

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To prove that a continuous periodic piecewise differentiable function \( f(\theta) \) can be expressed as \( f(\theta) = f(0) + \int_0^{\theta} g(t) dt \) for a piecewise continuous function \( g \), start by considering the differentiability of \( f \) except at a finite number of points. At points where \( f \) is differentiable, \( g(t) \) can be defined as the derivative \( f'(t) \). At the non-differentiable points, \( g(t) \) can still be defined in a piecewise manner, ensuring continuity. This approach leverages the periodic nature of \( f \) and the properties of integrals to establish the desired relationship. Thus, the proof hinges on the behavior of \( f \) and \( g \) around the points of non-differentiability.
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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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Can you do it when $f$ is differentiable? THen use the fact that there are only finite many points where $f$ is not differentiable.
 

Thread 'About the definition of topological manifold using closed sets'

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