The coefficients may be taken as real numbers, for real-valued functions.

Problem 5 f and g are real-valued functions defined on the real line.

Consider the space of real-valued functions together with a special point .

Let be a real-valued function of a real variable.

Let be an odd real-valued function on a circle.

As a real-valued function, often describes some numerical quantity of a given event.

For example, the `points' could consist of all real-valued functions differentiable on the whole real line.

Let f be a real-valued function defined on all of (a, b) except possibly at p itself.

We will merely note here that an observable is some real-valued function f on the state space.

Real-valued functions of several real variables appear pervasively in economics.

The following table contains a few particularly important types of real-valued functions:

A5. f is a strictly increasing real-valued function on the reals.

The limit of a real-valued function of several real variables is as follows.

For a real-valued function f(t), and so one can cut the number of independent expansion coefficients in half.

Note that although is usually a real-valued function (), it does not return a probability.

Let be real-valued functions defined on a set .

Indeed, this relation cannot cannot be represented by a continuous real-valued function.

This is thematically related to the superadditivity property of real-valued functions.

Suppose is a set and is a real-valued function for every natural number .

Differentiation and integration of real-valued functions of several real variables.

When referring to real-valued functions, usually the L2 inner product is assumed unless otherwise stated.

In order to make f a real-valued function, we must dispense with the final term .

Let f denote a real-valued function defined on a subset I of the real numbers.

For simplicity, in this article a real-valued function of several real variables will be simply called a function.

The set of all continuous real-valued functions defined on the real line forms a commutative ring.

My real function is to keep you from going crazy.

It has no real function other than to raise its members to power.

But I had no time to perform the real function of my position.

We already have more than enough ways to hit countries with nuclear weapons, and that is its only real function.

In fact, there are only four rooms that have a real function.

It is some form of service robot, though its real function I do not know.

Defence is the only real function of the government.

Suppose that some family of real functions on was chosen.

"The real function of teaching is to make yourself obsolete," he said.

"It doesn't seem to have any real function other than gender identification," she said.

I'm hearing about tendons and before I didn't know its real function.

Here, it is important to note that the real functions span the same space as the complex ones would.

This is a beginning step in accepting your real function on earth."

At least that comes closer to their real function - nourishment.

It could be interesting -educational-on the other hand, he had no real function there now.

"We didn't see any real function for it and it seemed the perfect place to put something in."

Its real function has always been to protect a tiny elite by intimidating the poor.

Indeed, until now, the cycles had no real function.

There seemed to be an impalpable wall around all normal, real functions in this house.

She doesn't seem to have any real function, Willie.

As they did, Harvath tried to figure out what the hell the operations real function was.

The pink cabinet, unlike the other cabinets, had a real function.

A question begins to arise: what real function do the archive and the committee play in these works?

It downsizes the real functions of government and the public sector even further.

Many important function spaces are defined to consist of real functions.