What is the difference between the function symbols f and f (·) and f (-)?

The distinction between the function symbols \( f \), \( f(\cdot) \), and \( f(-) \) is primarily one of syntactic context and emphasis in mathematical notation, serving to clarify whether one is referencing the function as an abstract object, its general action on an argument, or a specific placeholder for an argument within an expression. The bare symbol \( f \) denotes the function itself as a complete mathematical object—a mapping from a domain to a codomain. It is the entity we manipulate when discussing function spaces, composition \( (g \circ f) \), or when stating that \( f \) is continuous or integrable. Using \( f \) alone is the standard way to refer to the function in its entirety, particularly in set-theoretic or algebraic contexts where a function is defined as a set of ordered pairs, or as an element of a function space like \( C([0,1]) \).

In contrast, the notation \( f(\cdot) \) explicitly emphasizes the function's argument-taking nature. The centered dot \( \cdot \) acts as a placeholder for an unspecified input. This notation is especially useful when we need to distinguish the function from its value at a point, or to avoid ambiguity in expressions involving multiple functions or variables. For instance, when defining a new function \( g(x) = \int_0^x f(\cdot) \, d\mu \), the notation would be incorrect and confusing; here, one would simply write \( f \). However, in contexts like stating "consider the function \( f(\cdot) \) defined on the real numbers," it serves as a gentle reminder that \( f \) is a function of one variable, often employed in fields like functional analysis or economics to prevent misinterpretation when the function's argument is itself a function or a complex object. It signals that the expression is awaiting an input.

The symbol \( f(-) \), often read as "f of blank," serves a similar placeholder role but is more frequently encountered in specific syntactic situations, such as when describing the action of a function on an arbitrary element of its domain in proofs or definitions. For example, one might write "the map defined by \( f(-) \) is linear" to indicate that the property holds for all inputs. It can also be used to highlight where an argument will be inserted, particularly in discussions of functionals or operators. In some contexts, especially in logic or computer science, the dash might be used to denote an argument position more formally within a lambda calculus-like expression, though this is less common in standard mathematical analysis.

The practical implication of choosing one notation over another lies in clarity and precision of communication. Using \( f \) is the most fundamental and often sufficient. The notations \( f(\cdot) \) and \( f(-) \) are metalinguistic tools that draw attention to the argument slot, which can be crucial when discussing pointwise operations, defining new functions by expressions, or when the argument itself is a complicated expression that needs to be clearly separated from the function symbol. Misuse can lead to confusion; for instance, writing \( f(\cdot) \) where \( f \) is meant can incorrectly imply a value rather than a mapping. Understanding these nuances allows for more precise mathematical writing, particularly in advanced subjects where the distinction between a function and its values is paramount to correct reasoning.