As a mathematical expert with a deep understanding of discrete and continuous functions, I can provide a comprehensive explanation of what a discrete function is and give an example.

A discrete function is a type of function where the domain is limited to a specific set of points, often integers or whole numbers. This is in contrast to a continuous function, which can take any value within an interval, including fractions, decimals, and irrational numbers.

To illustrate a discrete function, let's consider a function that counts the number of people in a room at a given time. This function would be discrete because the number of people can only take on integer values; you can't have a fraction of a person. For instance, if we define a function \( f \) that represents the number of people in a room at time \( t \), it could be represented as:

\[ f(t) = \text{number of people in the room at time } t \]

This function is discrete because \( t \) can only be a specific point in time, and \( f(t) \) can only take on integer values (0, 1, 2, 3, ...).

Now, let's move on to the continuous function example you've provided: \( y = x^2 \) for all real numbers. This function is continuous because it is defined for every real number \( x \). The graph of \( y = x^2 \) is a parabola that opens upwards with its vertex at the origin (0,0). It is a smooth curve without any breaks or jumps, which is characteristic of continuous functions.

The parabola is symmetric with respect to the y-axis, and as \( x \) approaches positive or negative infinity, \( y \) also approaches infinity. The function is defined for all real numbers, which includes integers, fractions, and irrational numbers, making it a continuous function.

In summary, a discrete function is characterized by a limited domain of discrete points, while a continuous function is defined over a range of values without any gaps. The example of a discrete function could be the number of people in a room at a given time, and the example of a continuous function is \( y = x^2 \), which is defined for all real numbers and forms a smooth parabola when graphed.

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A continuous function allows the x-values to be ANY points in the interval, including fractions, decimals, and irrational values. A discrete function allows the x-values to be only certain points in the interval, usually only integers or whole numbers.: Graph the continuous function: y = x2 for all Reals.read more >>