A locally integrable function has a value for a portion or “slice” of the function, even if the integral is undefined as a whole. For example, the Heaviside function pictured above can’t be integrated as a whole, but it can be integrated in smaller localities.
Implications of Local Integrability
The condition of being a locally integrable function implies that the function’s indefinite integral is locally absolutely continuous, but while all continuous functions are locally integrable (Al-Omari, 2014), the converse isn’t true: a locally integrable function isn’t necessarily continuous (although in many cases they are).
In addition, the absolute value of a locally integrable function is locally integrable (Debnath & Mikusinski, 2005). However, just because a function is locally integrable doesn’t mean that the derivative there is also locally integrable (Park & Regensburger, 2011).
Formal Definition of Locally Integrable Function
More formally, a locally integrable function is defined as a bounded function (i.e. a function with and interval consisting of an upper bound or lower bound), rather than having infinite bounds (i.e. bounds of -∞, ∞). This can be written as follows:
“A Locally integrable function is a function f defined on ℝ such that for any compact interval [a, b], the integral exists” ~ David Hilbert.
Al-Omari, S. On a Widder potential transform and its extension to a space of locally integrable Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences Volume 18, 2015 – Issue 1.
Debnath, L. & Mikunsinski, P. (2005). Introduction to Hilbert Spaces with Applications 3rd Edition. Academic Press.
Hilbert, D. (2008). A quick look at topological and functionalspaces.
Park, H. & Regensburger, G. Gröbner Bases in Control Theory and Signal Processing. De Gruyter. 2011.
Turner, M. & Bates, D. (Eds.) Mathematical Methods for Robust and Nonlinear Control: EPSRC Summer School. Springer London. 2007.
Stephanie Glen. "Locally Integrable Function: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/locally-integrable-function/
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