Mathematics and Statistics · Class 11 Commerce
Ch 8Continuity — Class 11 Mathematics and Statistics, concept-first.
Informally, a function is continuous at a point if its graph can be drawn through that point without lifting the pen — no break, no hole, and no sudden jump. To turn this picture into a precise, testable rule, recall from the Limits chapter that a function's limit at a point describes the value the function approaches,…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Continuity At A Point
Imagine drawing the graph of a function and putting your pen down at . If the function is continuous there, you can draw straight through that point without lifting your pen — no jump, no hole, no break.
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Continuity of a Function at a Point
Informally, a function is continuous at a point if its graph can be drawn through that point without lifting the pen — no break, no hole, and no sudden jump.
Continuity from Left, from Right, and over an Interval
The single-point test of §1 extends naturally to one-sided continuity and to continuity across a whole stretch of the domain.
Types of Discontinuity
When a function fails the three-condition test of §1 at a point, the way it fails classifies the discontinuity. There are three standard types met at this level.
Continuity of Standard Functions
Rather than apply the three-condition test from scratch every time, it is far quicker to know which standard functions are continuous and where.
Algebra of Continuous Functions
Continuity is preserved under the ordinary arithmetic operations. This is what lets complicated functions be judged continuous by breaking them into simple, already-known continuous pieces.
Finding Constants that Make a Function Continuous
A very common problem type gives a function containing one or more unknown constants and asks for the value(s) that make the function continuous at a stated point (or across its whole domain).
Exercises
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- Q11Discuss the continuity of $f(x) = 2x^3 - x + 5$ at $x = 2$.Free
- Q12A function is defined by $f(x) = \dfrac{x^2 - 16}{x - 4}$ for $x \neq 4$ and $f(4) = 10$. Examine the continuity of $f$ at $x = 4$ and state…Free
- Q13Find the value of $k$ for which $f(x) = \dfrac{x^2 - 25}{x - 5}$ (for $x \neq 5$), $f(5) = k$, is continuous at $x = 5$.Preview
- Q14Find the values of $a$ and $b$ so that $f(x) = \begin{cases} 1, & x \le 0 \\ ax + b, & 0 < x < 1 \\ 5, & x \ge 1 \end{cases}$ is continuous…Preview
More questions
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- Example 1Examine the continuity of $f(x) = x^2 + 3x - 2$ at $x = 1$.Free
- Example 2A function is defined by $f(x) = \dfrac{x^2 - 4}{x - 2}$ for $x \neq 2$ and $f(2) = 5$. Examine the continuity of $f$ at $x = 2$, and if it…Free
- Example 3Examine the continuity at $x = 2$ of the function $f(x) = \begin{cases} x + 1, & x \le 2 \\ 3x - 1, & x > 2 \end{cases}$ and, if discontinuo…Free
- Example 4Examine the continuity of the modulus function $f(x) = |x - 2|$ at $x = 2$.Preview
- Example 5Discuss the continuity of $f(x) = \dfrac{1}{x - 3}$, and state the interval(s) on which it is continuous.Preview
- Example 6Find the value of $k$ for which $f(x) = \dfrac{x^2 - 9}{x - 3}$ (for $x \neq 3$), $f(3) = k$, is continuous at $x = 3$.Preview
- Example 7Find the value of $k$ so that $f(x) = \dfrac{\sqrt{x} - 2}{x - 4}$ (for $x \neq 4$), $f(4) = k$, is continuous at $x = 4$.Preview
- Example 8Find the values of $a$ and $b$ so that $f(x) = \begin{cases} 5x - 2, & x \le 1 \\ ax + b, & 1 < x < 3 \\ 7, & x \ge 3 \end{cases}$ is contin…Preview
- Example 9Show that $f(x) = x^2 + |x - 1|$ is continuous at every real number, using the algebra of continuous functions.Preview
- Example 10Find all points of discontinuity of $f(x) = \dfrac{x + 1}{x^2 - 5x + 6}$ and state the type of discontinuity at each.Preview