Elementary Functions

The Elementary Functions don’t do weird things. They are almost always continuous everywhere (in their domain). When they do have discontinuities or singularities, they are very nice with obvious jumps.

• Polynomial Function

  Polynomials are continuous everywhere. They have no discontinuities or singularities. Constant functions and linear functions are the nicest of the polynomial functions.

• Roots and Radicals

  Basic roots and radical functions are continuous everywhere. Not everywhere on the real numbers, because their domains are often not $\mathbb {R}$. Everywhere for functions means everywhere on their domain.

• Exponential Functions

  Basic exponential functions are continuous everywhere (on their domain).

• Logarithmic Functions

  Basic logarithmic functions are continuous everywhere (on their domain).

• Rational Functions

  Rational functions are continuous everywhere (on their domain). They have singularities and these are represented with vertical asymptotes and holes on their graphs. But singularities are not in the domain. Rational functions are not defined at singularities. Rational functions do not have discontinuities. They are continuous everywhere (on their domain).

• Absolute Value

  The basic absolute value function is continuous everywhere.

• Trigonometric Functions Trigonometric functions are continuous everywhere (on their domain). They have singularities, like tangent, and these singularities are represented with vertical asymptotes on the graphs. But singularities are not in the domain. The trigonometric functions are not defined at singularities. Trigonometric functions do not have discontinuities. They are continuous everywhere (on their domain).

The Elementary Functions are very nice. They have no discontinuites. They are continuous everywhere on their domains - or just continuous everywhere.

The first example of a simple function with a discontinuity is the Heaviside step (unit step) function.

$$Heaviside(x) = \begin {cases} 0 & \text { on } (-\infty , 0) \\ \tfrac {1}{2} & \text { at } 0 \\ 1 & \text { on } (0, \infty ) \end {cases}$$

We also have the Greatest Integer function - also called the floor function.

$$floor(x) = \lfloor x \rfloor = \, \text { the greatest integer less than or equal to } \, x$$

Graph of $y = \lfloor x\rfloor$ is below, except the steps keep going up to the right and down to the left. The extra 3 dots are a graphical symbol communicating that the pattern continues.

The floor function illustrates a general feeling about elementary functions. The only way discontinities are created is through the use of piecewise defined functions. To create a discontinuity, we have to break a nice function and just move a piece of it to somewhere else.

Forcing Discontinuities

Let’s make some discontinuities.

We’ll take pieces of different elementary functions and glue them together via piecewise defined functions.

Jumps

$$\text {Let } \, G(x) = \begin {cases} (x+6)(x+2) & \text { on } (-\infty , -3] \\ 2\sin (x) & \text { on } (-3, 4) \\ 2 & \text { at } 4 \\ 2(x-4)-3 & \text { on } (4, \infty ) \end {cases}$$

• $G$ is continuous on $(-\infty ,-3]$, because $G$ is a quadratic function on this interval.
• $G$ is continuous on $(-3,4]$, because $G$ is a sine function on this interval.
• $G$ is continuous on $(4,\infty )$, because $G$ is a linear function on this interval.

Basic elementary functions are continuous (on thier domains).

As we will see when studying compositions in more detail, if you compose a nice basic elementary function with a discontinuous function, then you get a discontinuous function. If you break the domain up into pieces, then the range will follow suit.