What is the proper term for a function which is NOT defined piecewise? I cannot seem to find a straight answer anywhere on the Internet, but I would really like to know the term for a function which is defined by a single expression over it's entire domain, e.g.
$$f(x)=x^{3}-x \quad\text{or}\quad g(x)=\sqrt{x},$$
as opposed to something like
$$f(x)
= \begin{cases}
x^3-x & \text{if $x<0$, and} \\
\sqrt{x} & \text{if $x\ge 0$.}
\end{cases}
$$
My knowledge of advanced mathematical terminology is limited, so I may have simply not understood some of the things I've read on the web. Is it an 'entire' function? 'Explicit'? 'Holomorphic'? None of them? I wish I knew.
 A: TL;DR
No such term exists.
Discussion
First off, there are several terms which you have used which are definitely incorrect:


*

*In complex analysis, a holomorphic function is a function which is analytic on its domain.  This means that at any point in its domain, the function is given by a power series which converges on a ball of positive radius.  Equivalently, the function is smooth (as derivatives of all order; this equivalence is a theorem, not a definition).  This definition has nothing to do with a function being expressed by a single formula.

*An entire function is a function which is holomorphic (in the sense described above) on the entire complex plane.  That is, an entire function is a function
$$ f : \mathbb{C} \to \mathbb{C}$$
which is complex differentiable at every point.  Again, this says nothing about the function being defined by a single formula.

*An explicit formula is typically used to express the idea that a function can be given by a relatively simple formula or expression, even if it is presented in a more complicated manner.  For example, the sequence of Fibonacci numbers can be though of as a function
$$ F : \mathbb{N} \to \mathbb{N}
\qquad\text{defined by the relation}\qquad F(0) = F(1) = 1, \ F(n+1) = F(n) + F(n-1). $$
This relation recursively defines the Fibonacci numbers.  However, there is an explicit formula, as well:
$$ F(n) = 
\frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n\sqrt{5}}.
$$
Another example:  the equation $y/x = 1$ implicitly defines $y$ as a function of $x$.  An explicit formula is given by $y(x) = x$ (where $x \ne 0$, since $y/x$ is undefined for this value of $x$).  It is worth noting that piecewise functions can be given explicitly.  For example, the absolute value function is explicitly given by
$$ |x| = \begin{cases}
x & \text{if $x\ge 0$, and} \\
-x & \text{if $x < 0$.}
\end{cases}
$$
This is both explicit and piecewise.  Once again, being expressible by a single formula or expression is a concept orthogonal to the notion of being explicitly expressed.

*Finally, since it has been brought up in the comments, there is a notion of an elementary function.  Elementary functions are those functions which can be written as the sum, product, and/or composition of


*

*algebraic functions (polynomials, rational functions, roots, etc);

*trigonometric functions (sine, cosine, tangent, etc) and their inverses;

*hyperbolic trigonometric functions and their inverses; and

*exponential functions and their inverses (note that this actually covers the trigonometric and hyperbolic trigonometric functions if we are comfortable working over the complex numbers).
While these functions are certainly good candidates for functions which are expressible by a single formula on their domains, I don't think that this category is large enough to encompass all such functions.  For example,


*

*The Gamma function, which generalizes the factorial function, can be defined by the integral
$$ \Gamma(s) = \int_{0}^{\infty} t^{s-1} \mathrm{e}^{-t}\ \mathrm{d}t $$
on the entire domain $s > 0$.  This is not an elementary function, but is expressed by a single formula.

*The error function, or $\operatorname{erf}$ function, which is useful in (for example) probability.  This function describes the area under the graph of a standard normal distribution between $-x$ and $x$.  It is given by
$$ \operatorname{erf}(x) = \frac{1}{\sqrt{\pi}} \int_{-x}^{x} \mathrm{e}^{-t^2},\mathrm{d} t. $$

*For $\alpha > 0$, the function $J_{\alpha}$ denotes the Bessel function of the first kind of order $\alpha$.  By definition, $J_{\alpha}$ solves the differential equation
$$ x^2 u_{xx} + x u_x + (x^2 - \alpha^2) u = 0. $$
It can be shown that $J_{\alpha}$ is explicitly given by the formula
$$
J_{\alpha}(x)
= \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left( \frac{x}{2} \right)^{2m+\alpha}
$$
For integer values of $\alpha$, this formula describes the function on the whole domain of $J_{\alpha}$, yet this function is far from elementary.
Finally getting to the question asked:  to my knowledge, there is no widely used and commonly understood term which describes a function which is expressed by a single formula on its domain.  Moreover, I do not think that such a term would be all that useful—there are simply too many ways of expressing functions, and the notion of a "single expression" is incredibly nebulous.  For example, as noted in the comments, the absolute value function can be written as a piecewise function, but it is also true that for any $x \in \mathbb{R}$,
$$
|x| = \sqrt{x^2}.
$$
Another set of examples are continuous periodic functions on $\mathbb{R}$.  Such functions have convergent Fourier series expansions, and can therefore be expressed by a single formula—their Fourier series.  In general, if $f: [-L,L] \to \mathbb{R}$ is continuous, then
$$ f(x) = \frac{a_0}{2} + \sum_{m=1}^{\infty} \left[ a_n \cos\left( \frac{m\pi x}{L} \right) + b_n \sin\left( \frac{m\pi x}{L} \right) \right], $$
where the Fourier coefficients are given by
$$
a_m = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left( \frac{m\pi x}{L} \right)\ \mathrm{d}x
\qquad\text{and}\qquad
b_m = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left( \frac{m\pi x}{L} \right)\ \mathrm{d}x
$$
So there is a large class of a priori piecewise defined functions which can be expressed by a "single formula".
If having a single formula over the entire domain is really a desirable property which needs to be discussed, I think that the best you can do is use plain English to describe this property.  For example, "We consider the class of functions which can be expressed by a single formula, rather than via a piecewise definition," or some such.
