Mathematics · Class 11 Science
Ch 17Continuity — Class 11 Mathematics, concept-first.
The word "continuity" in everyday language means an unbroken, consistent existence over a stretch of time or space: an unbroken road joining two cities, the steady flow of a river, an unbroken length of railway track, or the way a city's temperature changes gradually through the day rather than jumping about.
Key concepts
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Algebra of Continuous Functions
If two functions and are each already known to be continuous at a point , then several natural combinations built from them are automatically continuous there too, without any further checking: their sum , their differen…
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Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Continuous and Discontinuous Functions
The word "continuity" in everyday language means an unbroken, consistent existence over a stretch of time or space: an unbroken road joining two cities, the steady flow of a river, an unbroken length…
+−EXERCISE 8.143 questions
- Q1Examine the continuity of $f(x) = x^3 + 2x^2 - x - 2$ at $x = -2$.Free
- Q2Examine the continuity of $f(x) = \sin x$, for $x \le \dfrac{\pi}{4}$, $= \cos x$, for $x > \dfrac{\pi}{4}$, at $x = \dfrac{\pi}{4}$.Free
- Q3Examine the continuity of $f(x) = \dfrac{x^2-9}{x-3}$, for $x \ne 3$, $= 8$ for $x = 3$.Free
- Q4Examine whether the function is continuous at the point indicated against it: $f(x) = x^3 - 2x + 1$, if $x \le 2$, $= 3x - 2$, if $x > 2$, a…Preview
- Q5Examine whether the function is continuous at the point indicated against it: $f(x) = \dfrac{x^2+18x-19}{x-1}$, for $x \ne 1$, $= 20$ for $x…Preview
- Q6Examine whether the function is continuous at the point indicated against it: $f(x) = \dfrac{x}{\tan 3x} + 2$, for $x < 0$, $= \dfrac{7}{3}$…Preview
- Q7Find all the points of discontinuities of $f(x) = \lfloor x \rfloor$ on the interval $(-3, 2)$.Preview
- Q8Discuss the continuity of the function $f(x) = |2x + 3|$, at $x = -3/2$.Preview
- Q9Test the continuity of the following function at the point indicated against it: $f(x) = \dfrac{\sqrt{x-1} - (x-1)^{1/3}}{x-2}$, for $x \ne…Preview
- Q10Test the continuity of the following function at the point indicated against it: $f(x) = \dfrac{x^3-8}{\sqrt{x+2} - \sqrt{3x-2}}$, for $x \n…Preview
- Q11Test the continuity of the following function at the point indicated against it: $f(x) = 4x+1$, for $x \le 8/3$, $= \dfrac{59-9x}{3}$, for $…Preview
- Q12Test the continuity of the following function at the point indicated against it: $f(x) = \dfrac{(27-2x)^{1/3} - 3}{9 - 3(243+5x)^{1/5}}$, fo…Preview
- Q13Test the continuity of the following function at the point indicated against it: $f(x) = \dfrac{x^2+8x-20}{2x^2-9x+10}$, for $0 < x < 3$, $x…Preview
- Q14Identify discontinuities for the following function as either a jump or a removable discontinuity: $f(x) = \dfrac{x^2-10x+21}{x-7}$.Preview
- Q15Identify discontinuities for the following function as either a jump or a removable discontinuity: $f(x) = x^2+3x-2$, for $x \le 4$, $= 5x+3…Preview
- Q16Identify discontinuities for the following function as either a jump or a removable discontinuity: $f(x) = x^2-3x-2$, for $x < -3$, $= 3+8x$…Preview
- Q17Identify discontinuities for the following function as either a jump or a removable discontinuity: $f(x) = 4+\sin x$, for $x < \pi$, $= 3-\c…Preview
- Q18Show that the following function has continuous extension to the point where $f(x)$ is not defined. Also find the extension: $f(x) = \dfrac{…Preview
- Q19Show that the following function has continuous extension to the point where $f(x)$ is not defined. Also find the extension: $f(x) = \dfrac{…Preview
- Q20Show that the following function has continuous extension to the point where $f(x)$ is not defined. Also find the extension: $f(x) = \dfrac{…Preview
- Q21Discuss the continuity of the following function at the point indicated against it: $f(x) = \dfrac{\sqrt3 - \tan x}{\pi - 3x}$, $x \ne \dfra…Preview
- Q22Discuss the continuity of the following function at the point indicated against it: $f(x) = \dfrac{e^{1/x}-1}{e^{1/x}+1}$, for $x \ne 0$, $=…Preview
- Q23Discuss the continuity of the following function at the point indicated against it: $f(x) = \dfrac{4^x - 2^{x+1}+1}{1-\cos 2x}$, for $x \ne…Preview
- Q24Which of the following functions has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become…Preview
- Q25Which of the following functions has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become…Preview
- Q26Which of the following functions has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become…Preview
- Q27Which of the following functions has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become…Preview
- Q28Which of the following functions has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become…Preview
- Q29If $f(x) = \dfrac{\sqrt{2+\sin x} - \sqrt3}{\cos^2 x}$, for $x \ne \dfrac{\pi}{2}$, is continuous at $x = \dfrac{\pi}{2}$ then find $f\left(…Preview
- Q30If $f(x) = \dfrac{\cos^2 x - \sin^2 x - 1}{\sqrt{3x^2+1} - 1}$ for $x \ne 0$, is continuous at $x = 0$ then find $f(0)$.Preview
- Q31If $f(x) = \dfrac{4^{x-\pi} + 4^{\pi-x} - 2}{(x-\pi)^2}$ for $x \ne \pi$, is continuous at $x = \pi$, then find $f(\pi)$.Preview
- Q32If $f(x) = \dfrac{24^x-8^x-3^x+1}{12^x-4^x-3^x+1}$, for $x \ne 0$, $= k$, for $x = 0$, is continuous at $x = 0$, find $k$.Preview
- Q33If $f(x) = \dfrac{5^x+5^{-x}-2}{x^2}$, for $x \ne 0$, $= k$ for $x = 0$, is continuous at $x = 0$, find $k$.Preview
- Q34If $f(x) = \dfrac{\sin 2x}{5x} - a$, for $x > 0$, $= 4$ for $x = 0$, $= x^2+b-3$, for $x < 0$, is continuous at $x = 0$, find $a$ and $b$.Preview
- Q35For what values of $a$ and $b$ is the function $f(x) = ax + 2b + 18$, for $x \le 0$, $= x^2+3a-b$, for $0 < x \le 2$, $= 8x-2$, for $x > 2$,…Preview
- Q36For what values of $a$ and $b$ is the function $f(x) = \dfrac{x^2-4}{x-2}$, for $x < 2$, $= ax^2-bx+3$, for $2 \le x < 3$, $= 2x-a+b$, for $…Preview
- Q37Discuss the continuity of $f$ on its domain, where $f(x) = |x+1|$, for $-3 \le x \le 2$, $= |x-5|$, for $2 < x \le 7$.Preview
- Q38Discuss the continuity of $f(x)$ at $x = \dfrac{\pi}{4}$ where, $f(x) = \dfrac{(\sin x + \cos x)^3 - 2\sqrt2}{\sin 2x - 1}$, for $x \ne \dfr…Preview
- Q39Determine the values of $p$ and $q$ such that the following function is continuous on the entire real number line: $f(x) = x+1$, for $1 < x…Preview
- Q40Show that there is a root for the equation $2x^3 - x - 16 = 0$ between 2 and 3.Preview
- Q41Show that there is a root for the equation $x^3 - 3x = 0$ between 1 and 2.Preview
- Q42Activity: Let $f(x) = ax + b$ (where $a$ and $b$ are unknown), for $x < 1$, $= x^2 + 5$, for $x \ge 1$. Find the values of $a$ and $b$, so t…Preview
- Q43Activity: Suppose $f(x) = px + 3$ for $a \le x \le b$, $= 5x^2 - q$ for $b < x \le c$. Find the condition on $p, q$, so that $f(x)$ is conti…Preview
Continuity of a Function at a Point
To see what continuity should mean at a single point , it helps to first look at three graphs of that each fail to be continuous there in a different way (Fig. 8.1, Fig. 8.2, Fig.
Definition of Continuity
Putting together the three requirements noticed in 8.1.1, a function is said to be continuous at a point if all three of the following conditions hold:
Continuity from the Right and from the Left
When a function switches formula at a point, continuity there is checked one side at a time.
Examples of Continuous Functions
Rather than testing every new function from the three-condition definition each time, it is convenient to keep a short list of standard families that are already known to be continuous everywhere they…
Properties of Continuous Functions
If two functions and are each continuous at , then so are the following combinations, built from them:
Types of Discontinuities
Having seen examples of both continuous and discontinuous functions, it is useful to classify how a discontinuity happens, since different types call for different language and different remedies.
Jump Discontinuity
As seen already in Fig. 8.2, it can happen that both the left-hand limit and the right-hand limit of a function at exist as finite numbers, but the two numbers are different from each other — so the g…
Removable Discontinuity
Some discontinuities are much milder than a jump: the function's limit at the point does exist, but either the function was never given a value there, or it was assigned a value different from that li…
Infinite Discontinuity
A third, more dramatic way a function can fail to be continuous is by growing without any bound at all as approaches the point — there is then no finite height the graph is settling towards, from one…
Continuity Over an Interval
Continuity has so far been discussed only at a single point. The same idea extends naturally to a whole interval.
The Intermediate Value Theorem for Continuous Functions
Theorem (Intermediate Value Theorem). If is a continuous function on a closed interval , and is any value between and , then for some in .
Solved Examples
This section works through twelve fully solved examples and one guided activity, applying every idea from 8.1.1–8.1.11 to concrete functions.
More questions
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- Q44Select the correct answer from the given alternatives. $f(x) = \dfrac{2^{\cot x}-1}{\pi-2x}$, for $x \ne \dfrac{\pi}{2}$, $= \log\sqrt2$, fo…Free
- Q45Select the correct answer from the given alternatives. If $f(x) = \dfrac{1-\sqrt2\sin x}{\pi-4x}$, for $x \ne \dfrac{\pi}{4}$, is continuous…Free
- Q46Select the correct answer from the given alternatives. If $f(x) = \dfrac{(\sin 2x)\tan 5x}{(e^{2x}-1)^2}$, for $x \ne 0$, is continuous at $…Free
- Q47Select the correct answer from the given alternatives. $f(x) = \dfrac{x^2-7x+10}{x^2+2x-8}$, for $x \in [-6,-3]$. (A) $f$ is discontinuous a…Preview
- Q48Select the correct answer from the given alternatives. If $f(x) = ax^2+bx+1$, for $|x-1| \ge 3$ and $= 4x+5$, for $-2 < x < 4$, is continuou…Preview
- Q49Select the correct answer from the given alternatives. $f(x) = \dfrac{(16^x-1)(9^x-1)}{(27^x-1)(32^x-1)}$, for $x \ne 0$, $= k$, for $x=0$,…Preview
- Q50Select the correct answer from the given alternatives. $f(x) = \dfrac{32^x-8^x-4^x+1}{4^x-2^{x+1}+1}$, for $x \ne 0$, $= k$, for $x=0$, is c…Preview
- Q51Select the correct answer from the given alternatives. If $f(x) = \dfrac{12^x-4^x-3^x+1}{1-\cos 2x}$, for $x \ne 0$ is continuous at $x=0$ t…Preview
- Q52Select the correct answer from the given alternatives. If $f(x) = \left(\dfrac{4+5x}{4-7x}\right)^{4/x}$, for $x \ne 0$ and $f(0)=k$, is con…Preview
- Q53Select the correct answer from the given alternatives. If $f(x) = \lfloor x \rfloor$ for $x \in (-1,2)$ then $f$ is discontinuous at (A) $x=…Preview
- Q54Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = \dfrac{x^2-3x-10}{x-5}$, f…Preview
- Q55Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = 2x^2-2x+5$, for $0 \le x \…Preview
- Q56Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = \dfrac{\cos 4x - \cos 9x}{…Preview
- Q57Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = \dfrac{\sin^2 \pi x}{3(1-x…Preview
- Q58Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = \dfrac{|x+1|}{2x^2+x-1}$,…Preview
- Q59Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = \lfloor x+1 \rfloor$ for $…Preview
- Q60Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = 2x^2+x+1$, for $|x-3| \ge…Preview
- Q61Identify discontinuities if any for the following function as either a jump or a removable discontinuity on its domain: $f(x) = x^2+x-3$, fo…Preview
- Q62Identify discontinuities if any for the following function as either a jump or a removable discontinuity on its domain: $f(x) = x^2+5x+1$, f…Preview
- Q63Identify discontinuities if any for the following function as either a jump or a removable discontinuity on its domain: $f(x) = \dfrac{x^2+x…Preview
- Q64Discuss the continuity of the following function at the point or on the interval indicated against it. If discontinuous, identify the type o…Preview
- Q65Discuss the continuity of the following function at the point or on the interval indicated against it. If discontinuous, identify the type o…Preview
- Q66Find $k$ if the following function is continuous at the point indicated against it: $f(x) = \left(\dfrac{5x-8}{8-3x}\right)^{\frac{3}{2x-4}}…Preview
- Q67Find $k$ if the following function is continuous at the point indicated against it: $f(x) = \dfrac{45^x-9^x-5^x+1}{(k^x-1)(3^x-1)}$, for $x…Preview
- Q68Find $a$ and $b$ if the following function is continuous at the point indicated against it: $f(x) = \dfrac{4\tan x + 5\sin x}{a^x-1}$, for $…Preview
- Q69Find $a$ and $b$ if the following function is continuous on the interval indicated against it: $f(x) = ax^2+bx+1$, for $|2x-3| \ge 2$, $= 3x…Preview
- Q70Find $f(a)$, if $f$ is continuous at $x=a$ where, $f(x) = \dfrac{1+\cos(\pi x)}{\pi(1-x)^2}$, for $x \ne 1$ and at $a=1$.Preview
- Q71Find $f(a)$, if $f$ is continuous at $x=a$ where, $f(x) = \dfrac{1-\cos[7(x-\pi)]}{5(x-\pi)^2}$, for $x \ne \pi$ at $a = \pi$.Preview
- Q72Solve using the intermediate value theorem. Show that $5^x - 6x = 0$ has a root in $[1,2]$.Preview
- Q73Solve using the intermediate value theorem. Show that $x^3 - 5x^2 + 3x + 6 = 0$ has at least two real roots between $x=1$ and $x=5$.Preview