Question

What does it mean for a function to be continuous on an interval?

Short answer

Answer: A function is continuous on an interval if it meets all three conditions for continuity at every point within that interval, meaning there are no gaps, holes, or jumps in the function's graph between the interval's endpoints. This implies that the function's value and the limit of the function as x approaches any point within the interval are equal, and the function is defined at every point in the interval.

Step-by-step solution

Understand the concept of continuity

A function is continuous if its graph can be drawn without lifting the pencil, i.e., there are no "jumps" or "breaks" in the graph. Mathematically, this means the function has the property that for every point (x, f(x)) on its graph, the limit as x approaches that point equals the function's value at that point.

Provide the formal definition of continuity

In mathematical terms, a function, f(x), is continuous at a point x=a if the following three conditions are met: 1. f(a) is defined. 2. The limit of f(x) as x approaches a exists: \(\lim_{x\to a}f(x)=L\). 3. The limit of f(x) as x approaches a equals the function's value at a: \(\lim_{x\to a}f(x)=f(a)\).

Explain continuity on an interval

If a function is continuous at every point in an interval, we say it is continuous on that interval. This means that there are no gaps, holes, or jumps in the function's graph anywhere between the interval's endpoints.

Give an example of continuous and discontinuous functions

Examples: 1. Continuous function: A polynomial function, like f(x) = x^2, is continuous everywhere (on the entire real number line) because it meets all three conditions for continuity at every point. 2. Discontinuous function: The function g(x) = \(\frac{1}{x}\) is discontinuous at x=0 because the limit as x approaches 0 does not exist. However, g(x) is continuous on the intervals \((-\infty, 0)\) and \((0, \infty)\). By understanding the concept of continuity and the formal definition, you can determine whether a function is continuous on a given interval.