**Question:**
What does an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ look like locally?

As formulated, the question might be a bit difficult to answer since the Fourier transform of a function *f* ∈ *L*^{∞}(ℝ) is a distribution, and it is not easy to "write down" a distribution.
So let me first illustrate the situation at hand with an easy example:

The Fourier transform of the Heaviside functionExample:H(x) (i.e. the characteristic function of the positive reals) is given by a linear combination of the function 1/x and of the Dirac delta function (see this Wikipaedia entry for the exact formula, as well as for the meaning of the distribution "1/x").

The formalism of distributions is bit overkill for talking about measures, and things that look like 1/x. For example, the primitive of an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ is always a function (well defined outside of a set of measure zero). Using the above observation, we get the following

**Reformulation of the question:**

Let *f*(x) ∈ *L*^{∞}(ℝ) be a function, and let *g*(x) be a primitive of its Fourier transform.

• What can *g*(x) look like locally?

• What local conditions must *g* satisfy in order to have a chance of coming from some
*f* ∈ *L*^{∞}(ℝ)?

• On what kind of sets can *g* fail to be continuous?