SATHEE: Limit Continuity and Differentiability 5 Question 5

Limit Continuity and Differentiability 5 Question 5

5. Let $f : [- 1, 3] \to R$ be defined as

$f (x) = \begin{array}{cc} | x | + [x], & - 1 \leq x < 1 \\ x + | x |, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3, \end{array}$

(2019 Main, 8 April II) where, $[t]$ denotes the greatest integer less than or equal to $t$ . Then, $f$ is discontinuous at

(a) four or more points

(b) only two points

(d) only one point

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Answer:

Correct Answer: 5. (b)

Solution:

Since, $f (x)$ is continuous at $x = 0$ .

$\begin{array}{ll} \Rightarrow & lim_{x \to 0} f (x) = f (0) \\ \Rightarrow & f (0^{+}) = f (0^{-}) = f (0) = 0 \end{array}$

To show, continuous at $x = k$

$\begin{aligned} RHL = lim_{h \to 0} f (k + h) = lim_{h \to 0} [f (k) + f (h)] = f (k) + f (0^{+}) \\ = f (k) + f (0) \\ LHL = lim_{h \to 0} f (k - h) = lim_{h \to 0} [f (k) + f (- h)] \\ = f (k) + f (0^{-}) = f (k) + f (0) \\ ∴ lim_{x \to k} f (x) = f (k) \end{aligned}$

$\Rightarrow f (x)$ is continuous for all $x \in R$ .

Limit Continuity and Differentiability 5 Question 5

5. Let $f : [- 1, 3] \to R$ be defined as

Answer:

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Limit Continuity and Differentiability 5 Question 5

5. Let f:[−1,3]→R be defined as

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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5. Let $f : [- 1, 3] \to R$ be defined as