Question
Let $$f:R \to R$$ be a function defined as
\[f\left( x \right) = \left\{ \begin{array}{l}
5,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,\,\,\,x \le 1\\
a + bx,\,\,{\rm{if}}\,\,\,\,{\rm{1}} < x < 3\\
b + 5x,\,\,{\rm{if}}\,\,\,\,3 \le x < 5\\
30,\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,\,\,\,x \ge 5\,\,
\end{array} \right.\]
then, $$f$$ is-
A.
continuous if $$a=5$$ and $$b=5$$
B.
continuous if $$a =-5$$ and $$b= 10$$
C.
continuous if $$a=0$$ and $$b=5$$
D.
not continuous for any values of $$a$$ and $$b$$
Answer :
not continuous for any values of $$a$$ and $$b$$
Solution :
Let $$f\left( x \right)$$ is continuous at $$x = 1,$$ then
$$\eqalign{
& f\left( {{1^ - }} \right) = f\left( a \right) = f\left( {{1^ + }} \right) \cr
& \Rightarrow 5 = a + b......(a) \cr} $$
Let $$f\left( x \right)$$ is continuous at $$x = 3,$$ then
$$\eqalign{
& f\left( {{3^ - }} \right) = f\left( c \right) = f\left( {{3^ + }} \right) \cr
& \Rightarrow a + 3b = b + 15 \cr
& \Rightarrow a + 2b = 15......(b) \cr} $$
Solving (a) & (b) we get $$b= 10, \,\,a =-5$$
Now $$f\left( x \right)$$ is continuous at $$x = 5,$$ then
$$\eqalign{
& f\left( {{5^ - }} \right) = f\left( 5 \right) = f\left( {{5^ + }} \right) \cr
& \Rightarrow b + 25 = 30 \cr} $$
Which is not satisfied by $$a =-5$$ and $$b= 10.$$
Hence, $$f\left( x \right)$$ is not continuous for any values of $$a$$ and $$b$$