help with abstract math

Produced by Charles Wells.  Home    Website TOC    Website Index   Blog
Back to Functions Head 

Last edited 4/14/2009 10:23:00 AM



Functions may have names, for example "sine" or "the exponential function". The name in English and the symbol  for the function may be different; for example "sine" and "sin", "exponential function" and "exp".

A function may be named by a letter of the Roman or Greek alphabet, for temporary use in that particular section of text.  (This is what the Handbook calls a local identifier).  This usage is ubiquitous: 

¨  “Let F be the squaring function…” 

¨  “Let  be a continuous function with no derivative at 0.” 

¨  “Let  ”.  (  is the letter psi.)

Symbolic expressions that are not functions may also be given names (see here).


If  is the squaring function, then the name of the function (in this paragraph) is F.  The value of the function at 3 is F(3), which is 9.

Naming a function by its value at  x

It is common to refer to a function with identifier  (which may or may not be a name) as  (x) (of course some other variable may be used instead of  x). This is used with functions of more than one variable, too.


"Let h(x) be a continuous function."


"The function  is bounded."

Other ways of writing the value of a function are discussed under value of a function.

The defining expression as the name of a function

It is very common to refer to a function by using its formula.  This is common in calculus books.


"The derivative of  is always nonnegative."


“The derivative of  is always nonnegative.”  

If you analyze this latter example carefully, you see that it is literally nonsense.   The equation  is a statement.  How can a statement have a derivative?   Many mathematicians Frown Fiercely at this usage, but it is very common.

Barred arrow notation

Another technique is barred arrow notation.  If E is some mathematical expression that has a definite value for each x in the domain, then you can refer to the function  without having to give it a name. 


“The function  is not defined at 0.”  Here E is the expression . 


We could refer to the function G as “  ”.  For example, we could say:

“The derivative of  is  



It is also used with the plain arrow notation to define a function completely:

                                            “Let  ”.

Barred arrow notation may not be familiar to you, but it is becoming more common.  Like the defining expression, it allows you to refer to a function without giving it a name.   Using the barred arrow clears up ambiguity when the defining expression has parameters in it. 


Let .  This notation tells you that x is the function variable and y and z are parameters.  If you write “Consider the function  ” the sophisticated reader will assume you mean that x is the variable and a and b are parameters, by convention.  The barred arrow notation does not depend on knowledge of conventions.

Anonymous notation

Using barred arrow notation (“the function  ”) or defining expression (“the function  ”) to refer to a function are two examples of anonymous notation for functions.   Another anonymous notation used in theoretical computing science is lambda notation, where you would refer to the same function as .