Definite Integral of Function satisfying Dirichlet Conditions is Continuous

Theorem

Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$.

Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$.

Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$.

Then the real function:

$\map F x = \ds \int_{-\pi}^x \map f t \rd t - \dfrac {a_0} 2 x$

is continuous on $\openint {-\pi} \pi$.

Proof

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Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter Three: Properties of Fourier Series: $1$. Integration of Fourier Series