Determine the interval(s) on which the following functions are continuous. 
f(x)=1 / x^2−4

Verified step by step guidance

1. Identify the type of function: $f(x)=\frac{1}{x^2-4}$ is a rational function, which is continuous everywhere in its domain.
2. Determine where the denominator is zero: Set $x^2-4=0$ and solve for $x$.
3. Solve the equation: $x^2-4=0$ can be factored as $(x-2)(x+2)=0$, giving solutions $x=2$ and $x=-2$.
4. Identify the points of discontinuity: The function is discontinuous at $x=2$ and $x=-2$ because the denominator is zero at these points.
5. Determine the intervals of continuity: The function is continuous on the intervals $(-\mathrm{\infty },-2)$, $(-2,2)$, and $(2,\mathrm{\infty })$.

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This means there are no breaks, jumps, or asymptotes in the function's graph.

Identifying Discontinuities

Discontinuities in a function can occur at points where the function is undefined, such as division by zero. For the function f(x) = 1 / (x^2 - 4), we need to find values of x that make the denominator zero, as these points will indicate where the function is discontinuous. In this case, x^2 - 4 = 0 leads to x = ±2.

Intervals of Continuity

Once discontinuities are identified, the next step is to determine the intervals where the function remains continuous. For f(x) = 1 / (x^2 - 4), the function is continuous on the intervals that do not include the points of discontinuity. Thus, the intervals of continuity for this function are (-∞, -2) and (-2, 2) and (2, ∞).