Glossary

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous for all real numbers, as its graph flows smoothly without any breaks.

The function $f(x) = 1/(x-1)$ has a discontinuity at $x=1$ because the function is undefined there, leading to a vertical asymptote.

The domain of the function $f(x) = \sqrt{x-4}$ is $x \geq 4$, because the expression under the square root must be non-negative.

For the function $f(x) = x^2 + 5$, the function value at $x=2$ is $f(2) = 2^2 + 5 = 9$.

The greatest integer function, $f(x) = \lfloor x floor$, has a jump discontinuity at every integer value, as the function value abruptly changes.

For a piecewise function $f(x) = x+1$ for $x<2$, the left-hand limit as x approaches 2 is $2+1=3$.

For $f(x) = (x^2-9)/(x-3)$, the limit of the function as x approaches 3 is 6, even though the function is undefined at $x=3$.

The function $f(x) = (x^2-1)/(x-1)$ has a removable discontinuity at $x=1$ because it can be simplified to $x+1$ for $x eq 1$, creating a hole at $(1,2)$.

For a piecewise function $f(x) = 2x-1$ for $x \geq 2$, the right-hand limit as x approaches 2 is $2(2)-1=3$.

The function $f(x) = an(x)$ has vertical asymptotes at $x = \pi/2 + n\pi$ (where n is an integer), as the function's value approaches infinity or negative infinity at these points.