I am new to the field of distributions and reading some articles about them. I noticed, that it was written, that every linear bounded functional in $L^p$ can be written as $\displaystyle T(f)=\int f(x)g(x){\mathrm {d}}x$ with $g\in L^q, 1/p+1/q = 1$. (This follows from the Riesz representation theorem, right?)
Now the compactly supported smooth functions $C_c^\infty(\Omega),\Omega\subset \mathbb{R}^n$ are a subset of the $p$ integrable ones and on the former, distributions are defined as linear bounded functionals.
It is then mentioned, when introducing the dirac distribution, that this is not a function in the common sense, which is perfectly understandable.
They define it as $\displaystyle \int_{-\infty}^{\infty} f(x)\delta(x)\,\mathrm{d}x=f(0)$ where $f$ are compactly supported smooth functions.
But since this functional is linear and bounded, and $C_c^\infty(\Omega)\subset L^p(\Omega)$ and since the generating function is unique, this would mean, that $\delta$ is a function indeed.