A Complete Explanation of the Continuity over an Interval

A function is continuous over an interval when it is continuous at every point in it, with one-sided continuity at closed endpoints. Below: how the test differs for open, closed and half-open intervals, and the three kinds of discontinuity, removable, jump and infinite, that break it.

Understanding discontinuities and continuity over an interval is a fundamental concept in calculus and mathematical analysis. Let’s work through each of these topics in detail:

Continuity Over an Interval

Definition: A function is said to be continuous over an interval if it does not have any discontinuities within that interval. This means the function can be drawn without lifting the pen from the paper throughout the interval.

Types of Intervals:

1. Open Interval: An interval \( (a, b) \) is open if it includes all the points between \( a \) and \( b \) but not the endpoints \( a \) and \( b \) themselves.
2. Closed Interval: An interval \([a, b]\) is closed if it includes all the points between \( a \) and \( b \), including the endpoints.
3. Half-Open Interval: An interval that includes one endpoint but not the other, like \( [a, b) \) or \( (a, b] \).

Testing for Continuity:

• To determine if a function is continuous over an interval, one typically checks the continuity at each point in the interval.
• For closed intervals, one also needs to check the left-hand limit at the left endpoint and the right-hand limit at the right endpoint.

Examples:

• Polynomials: Polynomials are continuous over all real numbers, i.e., \((-\infty, \infty)\).
• Rational Functions: Functions like \( \frac{1}{x} \) are continuous everywhere except at points where the denominator is zero.

Discontinuities

Discontinuities occur where a function is not continuous. There are several types:

• Point (Removable) Discontinuity: Occurs if a function is not defined at a point, but the limit exists. For example,

\( f(x) = \frac{\sin(x)}{x} \) at \( x = 0 \).

This discontinuity can be “removed” by redefining the function at the point.

• Jump Discontinuity: This happens when the left and right limits exist at a point but are not equal. For example,

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\(f(x) = \begin{cases}
1, & \text{if } x > 0 \\
2, & \text{if } x \leq 0
\end{cases}\)

which jumps from one value to another. This Function is not continuous at \(x=0\)

• Infinite Discontinuity: Occurs when the function approaches infinity at a point. A common example is:

\( f(x) = \frac{1}{x} \) at \( x = 0 \).

• Oscillating Discontinuity: This is seen in functions that oscillate infinitely near a point, like:

\( f(x) = x\sin(\frac{1}{x}) \) at \( x = 0 \).

Identifying Discontinuities:

• Discontinuities can be identified by looking at the limits and values of a function. If a limit does not exist at a point, or if the limit differs from the function’s value, the function is discontinuous at that point.

Graphical Representation

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Graphs of functions are extremely helpful in visualizing continuity and discontinuity. Continuous functions show a smooth curve or line, while discontinuities might appear as breaks, jumps, or vertical asymptotes in the graph.

FAQ

What is continuity of a function on an interval?

A function is continuous on an interval when it is continuous at every point of that interval, so its graph has no holes, jumps or vertical asymptotes there. Informally, you can trace it from one end to the other without lifting your pencil. For example, f(x) = x² − 4x is continuous on any interval, because polynomials are continuous everywhere.

What does continuous on a closed interval mean?

A function is continuous on [a, b] if it is continuous at every point strictly between a and b and also one-sided continuous at the ends: the limit as x approaches a from the right equals f(a), and the limit as x approaches b from the left equals f(b). So f(x) = √x is continuous on [0, 4], even though it is not defined to the left of 0.

How do you check continuity on an interval?

Look for the points where a function can break: a zero in a denominator, a negative number under an even root, a log of a non-positive number, or the joining point of a piecewise function. If none of these fall inside the interval, the function is continuous there. For f(x) = 1/(x − 3), the interval (0, 2) is fine, but [0, 5] is not, since x = 3 lies inside it.

How do you prove continuity on an open interval?

Show that the function is continuous at every point c in (a, b), meaning the limit of f(x) as x → c equals f(c). Usually you do this by noting that f is built from continuous pieces, such as polynomials, sines or rational functions, combined by sums, products and quotients that never divide by 0 on the interval. An open interval needs no endpoint checks.

How do you determine the intervals of continuity of a function?

Find every x-value where the function is undefined or might jump, then split the number line at those points. For f(x) = (x + 1)/(x² − 4), the denominator is 0 at x = −2 and x = 2, so the function is continuous on (−∞, −2), (−2, 2) and (2, ∞). For a piecewise function, also test each boundary point with one-sided limits.

What is a closed interval?

A closed interval includes both of its endpoints and is written with square brackets, such as [1, 5], meaning all x with 1 ≤ x ≤ 5. An open interval like (1, 5) leaves out both endpoints, and a half-open interval like [1, 5) includes just one. The type matters for continuity, because each closed endpoint needs a one-sided check.