A function is called locally integrable if, around every point in the domain, there is a neighborhood on which the function is integrable. The space of locally integrable functions is denoted L_loc^1. Any integrable function is also locally integrable. One possibility for a nonintegrable function which is locally integrable is if it does not decay at infinity. For instance, f(x) = 1 is locally integrable on R, as is any continuous function.
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