Let f : R → R be a function defined as:

Question:

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined as :

$f(x)=\left\{\begin{array}{ccc}5, & \text { if } & x \leq 1 \\ a+b x, & \text { if } & 1

Then, $\mathrm{f}$ is :

  1. continuous if $a=5$ and $b=5$

  2. continuous if $a=-5$ and $b=10$

  3. continuous if $a=0$ and $b=5$

  4. not continuous for any values of $a$ and $b$


Correct Option: , 4

Solution:

$f(x)=\left\{\begin{array}{ccc}5 & \text { if } & x \leq 1 \\ a+b x & \text { if } & 1

$f(1)=5, \quad f\left(1^{-}\right)=5, \quad f\left(1^{+}\right)=a+b$

$\mathrm{f}\left(3^{-}\right)=\mathrm{a}+3 \mathrm{~b}, \mathrm{f}(3)=\mathrm{b}+15, \mathrm{f}\left(3^{+}\right)=\mathrm{b}+15$

$\mathrm{f}\left(5^{-}\right)=\mathrm{b}+25 ; \mathrm{f}(5)=30 \quad \mathrm{f}\left(5^{+}\right)=30$

from above we concluded that $f$ is not continuous for any values of $a$ and $b$.

Leave a comment

None
Free Study Material