Understanding Continuity in an Interval

Continuity on an interval is a foundational concept in calculus, crucial for understanding the behavior of real-valued functions across domains. This topic examines how functions remain unbroken, with no jumps, holes, or asymptotes within specific intervals. This article systematically presents definitions, formal criteria, properties, special cases, and rigorous solved examples relevant for JEE Main Mathematics.

Defining Continuity on an Interval

A function $f(x)$ is said to be continuous at a point $c$ if $\lim\limits_{x \to c} f(x) = f(c)$. Extending this notion, $f(x)$ is said to be continuous on an interval $I$ if it is continuous at every point $c \in I$.

More formally, a function $f$ is continuous on $I$ if for all $c \in I$, the following condition holds: $\displaystyle \lim_{x \to c} f(x) = f(c)$.

The definition of interval continuity encompasses both open and closed intervals, with endpoint behaviors treated distinctly in closed intervals.

Continuity on Open and Closed Intervals

On an open interval $(a, b)$, function $f(x)$ is continuous if $f$ is continuous at every $c$ such that $a < c < b$. There are no requirements for limits at the endpoints, since $a$ and $b$ are not included.

On a closed interval $[a, b]$, continuity at endpoints requires special attention:

• At $a$, $\displaystyle\lim_{x \to a^+} f(x) = f(a)$
• At $b$, $\displaystyle\lim_{x \to b^-} f(x) = f(b)$
• For $c \in (a, b)$, $f$ is continuous at $c$ as usual

Thus, continuity on $[a, b]$ demands continuity at each interior point, as well as right continuity at $a$ and left continuity at $b$.

Testing Continuity Across an Interval

To determine if a function $f(x)$ is continuous on an interval $I$, first establish the domain of definition. Then, at each point $c$ in $I$, verify that: $\displaystyle\lim_{x \to c^-} f(x)$, $\displaystyle\lim_{x \to c^+} f(x)$, and $f(c)$ all exist and are equal where applicable.

For closed intervals, always examine the behavior near the endpoints using one-sided limits. For open intervals, interior point continuity suffices.

Graphs can be a visual aid; the function should appear as a single unbroken curve without jumps, holes, or asymptotes on $I$. For detailed criteria and further explanations, consult Limit, Continuity, And Differentiability.

Key Properties of Continuous Functions on Intervals

Several important algebraic and structural properties hold for continuous functions:

• The sum, difference, and product of continuous functions are continuous on their common domains.
• The quotient $f(x)/g(x)$ is continuous wherever $g(x) \neq 0$.
• The composition $f \circ g$ is continuous wherever both functions are continuous and composition is defined.
• Polynomials, exponential functions, and trigonometric functions (in their respective domains) are continuous everywhere they are defined.

These properties facilitate the construction and analysis of complex continuous functions from simpler building blocks.

Formal Criteria and Mathematical Expression

For an open interval $(a, b)$: $f(x)$ is continuous on $(a, b)$ $\iff$ for every $c \in (a, b)$, $\displaystyle\lim_{x \to c} f(x) = f(c)$.

For a closed interval $[a, b]$:

• $\displaystyle\lim_{x \to a^+} f(x) = f(a)$
• $\displaystyle\lim_{x \to b^-} f(x) = f(b)$
• $f$ continuous on $(a, b)$

These conditions distinctly ensure that both interior points and endpoints comply with the requirements for interval continuity.

Examples of Continuity on Specific Intervals

Consider the function $f(x) = x^2$ on $[1, 3]$: - $f(x)$ is a polynomial, hence continuous everywhere on $\mathbb{R}$. - For $x = 1$: $\displaystyle\lim_{x \to 1^+} x^2 = 1^2 = 1 = f(1)$. - For $x = 3$: $\displaystyle\lim_{x \to 3^-} x^2 = 9 = f(3)$.

Thus, $f(x)$ is continuous on $[1, 3]$ by fulfilling all closed interval conditions. Explore more solved cases in Limits Solved Examples.

Example: Consider $f(x) = \dfrac{1}{x-2}$ on $[1, 3]$. Here, $f(x)$ is undefined at $x = 2$, which lies within the interval. Therefore, $f(x)$ is not continuous on $[1, 3]$ as the domain does not include the entire interval.

Example: $f(x) = \ln(x - 1)$ is continuous on $(1, 5]$ because its domain is $x > 1$, and for $x$ in $(1, 5]$ the function is defined and continuous everywhere on that set.

Distinguishing between Open and Closed Interval Continuity

Continuity on $(a, b)$ requires no conditions at the endpoints. On $[a, b]$, conditions at $a$ and $b$ must be checked via right- and left-hand limits respectively, in addition to interior points. This distinction is essential in problems where discontinuity may occur precisely at endpoints.

A summary of requirements:

Procedure: Checking Continuity on a Graph

Detecting intervals of continuity from a graph involves tracing the curve within the interval and identifying any location where abrupt jumps, holes, or asymptotes occur. Any such feature signals the absence of continuity at that point.

Regions where the function is smooth and connected correspond to intervals of continuity. This graphical approach supports, but does not replace, the rigorous analytical tests detailed earlier.

Note on Continuity and Discontinuity

A function may be continuous on some intervals and discontinuous on others, depending on its domain and the existence of discontinuities such as jumps, holes, or vertical asymptotes within a given interval.

Common discontinuities can be classified as removable (hole), jump, or infinite (asymptote), each violating the strict definition of continuity at certain points.

Solved Examples: Continuity on Intervals

Example 1: $f(x) = x^3 - 2x + 1$ on $[0, 4]$. Since $f(x)$ is a polynomial, it is continuous everywhere, including on $[0, 4]$.

Example 2: $f(x) = \dfrac{x - 1}{x^2 - 1}$ on $(1, 3)$. Simplify: $f(x) = \dfrac{1}{x + 1}$ for $x \neq 1$. Since $x \neq 1$ in $(1, 3)$, the function is continuous on $(1, 3)$.

Example 3: $f(x) = \sqrt{x}$ on $[0, 4]$. The function is defined and continuous over $[0, 4]$ since $\sqrt{x}$ is continuous for all $x \geq 0$.

Example 4: $f(x) = \dfrac{1}{x}$ on $(1, 5)$. The function is continuous everywhere except at $x = 0$, which is not in $(1, 5)$, so $f(x)$ is continuous on $(1, 5)$.

Example 5: $f(x) = \sin x$ on $[0, \pi]$. The sine function is continuous everywhere on $\mathbb{R}$, thus it is continuous on any closed interval such as $[0, \pi]$.

For more applications involving differentiability and continuity, refer to Differential Calculus.

Summary of Algebraic Properties of Continuous Functions

• The sum and product of continuous functions are continuous on their common domain.
• The quotient is continuous wherever the denominator is nonzero.
• The composition of two continuous functions is continuous wherever defined.
• Every polynomial, exponential, sine, and cosine function is continuous everywhere in its natural domain.

Practice Questions on Interval Continuity

1. Is $f(x) = e^{x}$ continuous on $[0, 2]$?
2. Determine whether $f(x) = \dfrac{1}{x-3}$ is continuous on $(2, 5)$.
3. Find intervals of continuity for $f(x) = \ln(x^2 - 4)$.

These questions cultivate analytical skills for rigorous testing of interval continuity, essential for advanced calculus and competitive examinations like JEE Main.

Further Exploration and Related Topics

Understanding continuity on intervals directly supports concepts such as the Intermediate Value Theorem and Mean Value Theorems. For detailed theory, solved problems, and practice resources relevant to JEE, see Limit Of A Function and Monotonicity And Extremum Of Functions.