Can distributions be thought of as functions of a real variable? I understand that, given some function space, distributions lie in the dual space.
In that sense, they can be thought of as functions of a "function of a real variable" variable.
But the common representation of the delta distribution as an infinitely large "spike" of a certain infinite magnitude suggests that perhaps they can also be thought of as functions of a real variable, where the codomain is some expanded set of which the reals are a subset.
Is this possible, or is delta a special case?
 A: The derivative of the Dirac delta will give you some trouble...
A: To complement the other answers: in many regards distributions can be thought of as "functions of a real variable", yes. (That's why I like Gelfand-etal's name "generalized functions" better. If it were just some dual space, it would not have the same usefulness...) But, yes, some cautions are necessary (as are necessary in other more conventional parts of real analysis, too!) Namely, distributions are indeed reliably thought-of as limits (in the weak dual topology) of very nice functions (e.g., smooth, compactly supported). But, then, unlike sup-norm limits, weak-dual limits do not say anything about pointwise limits, so, in particular, distributions do not reliably have pointwise values. But, after all, many more-familiar functions do not have pointwise values, either. E.g., an otherwise smooth function with a jump, but with left and right limits existing, really has no canonical value at the jump (despite occasional attempts to declare the average the correct value). 
That is, unlike "dual spaces" in general, the original space (of test functions) imbeds in its dual, and is dense. So distributions are an extension of "classical" functions, in many useful regards. It's just that the topology in which the limits are taken is pretty weak.
But, after all, if we take sup-norm limits of test functions, we typically lose all the differentiability, only retaining the continuity, so "lossy" limits are all around us.
A: The common representation that you mention is a non rigorous description that originates in the work of Physicists, before the precise mathematical theory of distributions had been developed. This theory does not treat
the delta distribution as a "function", and the fact that it is nonetheless called by that name (that is, the so-called "Dirac Delta Function") is just a historical consequence of the above. The Delta distribution need not be thought of as a function on $R$ or on an extended domain, but as a linear functional on the space of infinitely differentiable functions with compact support on the real line. This is the function-space whose dual consists of the distributions. 
There is a special case when a distribution may be treated as a function, and that is the case when a linear functional on the function-space is given by 
$$\Lambda_f(g)=\int_R f(x)g(x)\,dx$$
where $g$ is any function in $C_c^{\infty}(R)$ (the subscript denotes compact support), and $f$ is an integrable function (for example). In that case the distribution $\Lambda_f$ can, ought and should be identified with $f$.
On the other hand, the Delta distribution concentrated at some point $x_0$, is the functional given by
$$\Lambda(g)=g(x_0)$$
and there is no corresponding $\it{function}$  $f$ that will make this functional look like $\Lambda_f$ above.
