 Hi and welcome to the session. My name is Shashi and I am going to help you with the following question. Question says, discuss the continuity of the function f, where f is defined by fx is equal to minus 2. If fx is less than equal to minus 1, fx is equal to 2x. If x is greater than minus 1 and less than equal to 1, fx is equal to 2 if x is greater than 1. First of all let us understand that function f is continuous at point x is equal to a if f a exists or we can say the function is defined at x is equal to a. Second condition for the function to be continuous is value of the function is equal to right hand side limit of the function at x is equal to a that is f a is equal to limit of x tending to a plus fx is equal to limit of x tending to a minus fx. This is the key idea to solve the keyword question. Let us now start the solution. We are given fx is equal to minus 2 if x is less than equal to minus 1, fx is equal to 2x if x is greater than minus 1 and less than equal to 1, fx is equal to 2 if x is greater than 1. First of all let us consider the function fx is equal to minus 2. Clearly we can say this is a constant function. We know constant function is continuous at every real number so this implies function f is continuous at every real number less than minus 1. Another function fx is equal to 2x this is a polynomial function and we know polynomial function is continuous at every real number. So this implies function f is continuous at every real number between minus 1 and 1. Let us consider fx is equal to 2 again this is a constant function and we know constant function is continuous at every real number. This implies function f is continuous at every real number greater than 1. Let us take the continuity of the function at x is equal to minus 1. Clearly we can see the function is defined at x is equal to minus 1. We can write that x is equal to minus 1, function f is defined and right inside limit of the function at x is equal to minus 1. So we can write limit of x tending to minus 1 plus fx is equal to limit of x tending to minus 1 plus 2x this is equal to 2 multiplied by minus 1 which is further equal to minus 2. So right inside limit of the function at x is equal to minus 1 is equal to minus 2. Let us now find out left inside limit of the function at x is equal to minus 1. So we can write limit of x tending to minus 1 minus fx is equal to limit of x tending to minus 1 minus minus 2 which is further equal to minus 2. Clearly we can see the two limits coincide each other. Now let us find out value of the function at x is equal to minus 1. We know f minus 1 is equal to minus 2. So we can write value of the function is equal to left inside limit of the function is equal to right inside limit of the function. This implies function f is continuous at x is equal to minus 1. Let us now check the continuity of the function at x is equal to 1. Clearly we can see function is defined at x is equal to 1. So we can write fx is equal to 1. Option f is defined. Let us now find out right inside limit of the function at x is equal to 1. So we can write limit of x tending to 1 plus fx is equal to limit of x tending to 1 plus which is equal to 2. Let us now find out left inside limit of the function at x is equal to 1. That is limit of x tending to 1 minus fx is equal to limit of x tending to 1 minus 2x which is equal to 2 multiplied by 1 equal to 2. So we get left inside limit of the function at x is equal to 1 as 2. Now clearly we can see the two limits coincide each other. Now let us find out the value of the function at x is equal to 1. You know f1 is equal to 2, fx is equal to 2x or x is equal to 1. So we get the function is equal to value of the function at x is equal to 1. Or we can say right inside limit of the function is equal to left inside limit of the function is equal to value of the function at x is equal to 1. All of them are equal to 2. Now this implies function f is continuous at x is equal to 1. This means function f is continuous at every point in its domain and hence function f is a continuous function. This is our required answer. This completes the session. Hope you understood the session. Goodbye.