Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.
$h\left(x\right)=\frac{2x}{x^3-25x}$

Verified step by step guidance

1. Step 1: Identify the function given: \( h(x) = \frac{2x}{x^3 - 25x} \). This is a rational function, which is continuous everywhere in its domain.
2. Step 2: Determine the domain of the function by finding where the denominator is zero. Set \( x^3 - 25x = 0 \) and solve for \( x \).
3. Step 3: Factor the equation \( x^3 - 25x = x(x^2 - 25) = x(x - 5)(x + 5) = 0 \). The solutions are \( x = 0, x = 5, \) and \( x = -5 \). These are the points where the function is not defined.
4. Step 4: The function is continuous on the intervals where the denominator is not zero. These intervals are \((-\infty, -5)\), \((-5, 0)\), \((0, 5)\), and \((5, \infty)\).
5. Step 5: At the finite endpoints \( x = -5, 0, \) and \( 5 \), the function is not continuous. Therefore, there is no right- or left-continuity at these points.

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 concept is crucial for determining where a function does not have breaks, jumps, or asymptotes.

Identifying Discontinuities

Discontinuities can occur due to points where the function is undefined, such as division by zero, or where the left-hand limit and right-hand limit do not match. In the given function, h(x) = 2x / (x^3 - 25x), we need to find values of x that make the denominator zero, as these will indicate potential points of discontinuity.

Endpoints and One-Sided Limits

When analyzing continuity at finite endpoints of an interval, it is important to consider one-sided limits. A function can be left-continuous or right-continuous at an endpoint, meaning it approaches the endpoint from the left or right, respectively. This distinction is essential for accurately describing the behavior of the function at the boundaries of its domain.