Locally Integrable Function - Examples

Examples

  • The constant function 1 defined on the real line is locally integrable but not globally integrable. More generally, constants, continuous functions and integrable functions are locally integrable.
  • The function
f(x)= \begin{cases} 1/x &x\neq 0\\ 0 & x=0 \end{cases}
is not locally integrable in x = 0: it is indeed locally integrable near this point since its integral over any compact set not including it is finite. Formally speaking, 1/x ∈ L1,loc(ℝ\0).: however, this function can be extended to a distribution on the whole ℝ as a Cauchy principal value.
  • The preceding example raises a question: does every function which is locally integrable in Ω ⊊ ℝ admit an extension to the whole ℝ as a distribution? The answer is negative, and a counterexample is provided by the following function:
f(x)= \begin{cases} e^{1/x} &x\neq 0\\ 0 & x=0 \end{cases}
does not define any distribution on ℝ.
  • The following example, similar to the preceding one, is a function belonging to L1,loc(ℝ\0) which serves as an elementary counterexample in the application of the theory of distributions to differential operators with irregular singular coefficients:
f(x)= \begin{cases} k_1 e^{1/x^2} &x>0\\ 0 & x=0\\ k_2 e^{1/x^2} &x<0 \end{cases},
where k1 and k2 are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order
Again it does not defines any distribution on the whole ℝ, if k1 or k2 are not zero: the only distributional global solution of such equation is therefore the zero distribution, and this shows how, in this branch of the theory of differential equations, the methods of the theory of distributions cannot be expected to have the same success achieved in other branches of the same theory, notably in the theory of linear differential equations with constant coefficients.

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