 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 2x if x is less than 0, fx is equal to 0. If x is greater than equal to 0 and less than equal to 1, fx is equal to 4x if x is greater than 1. First of all let us understand that function f is continuous that x is equal to a if it is defined that x is equal to a or we can say f a exist. Then value of the function is equal to right hand side limit of the function is equal to left hand side limit of the function at x is equal to a. These are the two conditions which make that function continuous. This is the key idea to solve the given question. Let us now start the solution. We are given fx is equal to 2x if x is less than 0, fx is equal to 0. If x is greater than equal to 0 and less than equal to 1, fx is equal to 4x if x is greater than 1. First of all let us consider the function fx is equal to 2x. You can see this is the polynomial function and we know polynomial function is continuous at every real number. This implies function f is continuous at all real numbers less than 0. Let us consider the function fx is equal to 0. This is the constant function and we know constant function is continuous at every real number. This implies function f is continuous at every real number between 0 and 1. Let us now consider the function fx is equal to 4x. This is the polynomial function and we know polynomial function is continuous at all real numbers. This implies function f is continuous at every real number greater than 1. Now let us take the continuity of the function at x is equal to 0. Clearly we can see function f is defined at x is equal to 0. So we can write at x is equal to 0, function f is defined. Now let us find out right hand side limit of the function at x is equal to 0. So we can write limit of x tending to 0 plus fx is equal to limit of x tending to 0 plus 0 which is equal to 0. Now let us find out the left hand side limit of the function at x is equal to 0. So we get limit of x tending to 0 minus fx is equal to limit of x tending to 0 minus 2x which is equal to 0. We know 2 divided by 0 is equal to 0. Now let us find out the value of the function at x is equal to 0. We know f0 given to us is 0 only. Clearly we can see the 2 limits coincide each other and the value of the function at x is equal to 0. So we can write therefore limit of x tending to 0 plus fx is equal to limit of x tending to 0 minus fx is equal to f0 is equal to 0. This implies function f is continuous at x is equal to 0. Let us now check the continuity of the function at x is equal to 1. Clearly we can see the function is defined at x is equal to 1. So we can write at x is equal to 1. Function f is defined. Let us now find out right hand side 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 4x which is equal to 4 multiplied by 1 equal to 4. So we get limit of x tending to 1 plus fx is equal to 4. Let us now find left hand side limit of the function at x is equal to 1 is equal to limit of x tending to 1 minus 0 which is equal to 0 only. Now clearly we can see right hand side limit do not coincide left hand side limit. So we can write limit of x tending to 1 plus fx is not equal to limit of x tending to 1 minus fx 4 is not equal to 0. Since two limits do not coincide each other function f is not continuous at x is equal to 1. So the given function is continuous at all the points except for x is equal to 1. So we can write x is equal to 1 is the only point of discontinuity. So this is our required answer. This completes the session. Hope you understood the session. Goodbye.