 Hello and welcome to the session, I am Shashri and I am going to help you with the following question. Question says, discuss the continuity of the function F, where function F is defined by fx is equal to 3 if x is greater than equal to 0 and less than equal to 1, fx is equal to 4 if x is greater than 1 and less than 3, fx is equal to 5 if x is greater than equal to 3 and less than equal to 10. First of all let us understand that function F is continuous at x is equal to a if function is defined at x is equal to a or we can say f a exist and left hand side limit of the function is equal to right hand side limit of the function is equal to value of the function at x is equal to a. This is the key idea to solve the given question. Let us start with the solution now we are given fx is equal to 3 if x is greater than equal to 0 and less than equal to 1, fx is equal to 4 if x is greater than 1 and less than 3, fx is equal to 5 if x is greater than equal to 3 and less than equal to 10. Now clearly we can see function F is defined at all the real numbers from 0 to 10. So we can write function F is defined at all the real numbers 0 to 10, let us now discuss the continuity of the function for all the real values of x between 0 and 1 we know fx is equal to 3 if x is greater than equal to 0 and less than equal to 1. Now clearly we can see this is a constant function and constant function is continuous at every real number. So this implies function F is continuous at all the real numbers between 0 and 1. Now let us discuss continuity of the function for all the real values between 1 and 3 we know fx is equal to 4 if x is greater than 1 and less than 3. Now this is a constant function and we know constant function is continuous at every real number. So given function F is continuous at all the real numbers between 1 and 3 we are also given that fx is equal to 5 if x is greater than equal to 3 and less than equal to 10. This is again a constant function and constant function is continuous at every real number. So given function F is continuous at all the real numbers between 3 and 10. Now let us check continuity of the function at x is equal to 1. First of all let us find out right hand side limit of the function at x is equal to 1. This is given by limit of x tending to 1 plus fx which is equal to limit of x tending to 1 plus 4 which is equal to 4 we know fx is equal to 4 if x is greater than 1. Now let us find out left hand side limit of the function at x is equal to 1 that is equal to limit of x tending to 1 minus fx which is further equal to limit of x tending to 1 minus 3 which is further equal to 3 only we know for x less than 1 fx is equal to 3. Now clearly we can see right hand side limit is not equal to left hand side limit of the function at x is equal to 1. So we can write therefore limit of x tending to 1 plus fx is not equal to limit of x tending to 1 minus fx this implies function F is discontinuous at x is equal to 1. Now let us discuss continuity of the function at x is equal to 3 at x is equal to 3 right hand side limit is given by limit of x tending to 3 plus fx which is further equal to limit of x tending to 3 plus 5 which is equal to 5 we know for x greater than 3 fx is equal to 5. Now let us find out left hand side limit of the function at x is equal to 3 this is given by limit of x tending to 3 minus fx which is further equal to limit of x tending to 3 minus 4 we know for x less than 3 fx is equal to 4. Now this limit is equal to 4 now these two limits do not coincide each other so we can write limit of x tending to 3 plus fx is not equal to limit of x tending to 3 minus fx now this implies given function F is not continuous at x is equal to 3. So our required answer is given function F is not continuous at x is equal to 1 and x is equal to 3 this completes the session hope you understood the solution take care and have a nice day.