Question

f(x) = $\{\begin{array}{c}3,if0\le x<1\\ 4,if1<x<3\\ 5,if3\le x\le 10\end{array}$

Answer 1

Here, f(x) = $\{\begin{array}{c}3,if0\le x<1\\ 4,if1<x<3\\ 5,if3\le x\le 10\end{array}$

for x > 1, f(x) = x + 1 and x < 1, f(x) = $x^2$ + 1 iare constant funtions, so it is continuous in the above interval. so we have to check the continuity at x = 1,3.

At x=1, LHL = $\underset{x\to 1^{-}}{lim}$ f(x) = $\underset{x\to 1^{-}}{lim}$ (3) = 3, RHL = $\underset{x\to 1^{+}}{lim}$ f(x) = $\underset{x\to 1^{+}}{lim}$ (4) = 4

LHL $\ne$ RHL.

Thus, f(x) is discontinuous at x=1.

At x=1, LHL = $\underset{x\to 3^{-}}{lim}$ f(x) = $\underset{x\to 3^{-}}{lim}$ (4) = 4, RHL = $\underset{x\to 1^{+}}{lim}$ f(x) = $\underset{x\to 3^{+}}{lim}$ (5) = 5

$\therefore$ LHL $\ne$ RHL. Thus, f(x) is continuous everywhere except at x=1,3