Answer to Question #95325 in Real Analysis for Rajni

Question

Check whether the following function is continuous or not:
f(x)={6x for 0<=x>5 , 7 for x= 5, 3x+15 for 5<x<=10}
Also determine the type of discontinuity, if it exits?

Answer 1

First we check interval 0<=x<5, where function approaches 30 as x approaches 5 from negative side

Next, we check interval 5 < x <= 10, where function approaches same value (30) as x approaches 5 from positive side

Here, one-sided limit from the negative direction equals one-sided limit from the positive direction, but at the point x = 5, function equals to 7 ≠ 30. Therefore, this is a point with removable discontinuity.

Answer: f(x) is discontinuous, and the type of discontinuity is removable discontinuity.