 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, is the function f defined by fx is equal to x if x is less than equal to 1 and fx is equal to 5 if x is greater than 1. Continuous at x is equal to 0 and x is equal to 1 and x is equal to 2. 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. 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 now start with the solution we know fx is equal to x if x is less than equal to 1 and fx is equal to 5 if x is greater than 1. Now clearly we can see function f is defined at all the values of x less than 1 greater than 1 and equal to 1. So we can say function f is defined at all the real numbers. Let us now check the continuity of the function at x is equal to 0. We know fx is equal to x if x is less than equal to 1. So at x is equal to 0 fx is equal to x. So first of all we will find value of the function at x is equal to 0. So we get f0 is equal to 0. Now we will find limit of the function at x is equal to 0. First of all let us find out right hand side limit at x is equal to 0. This is equal to limit of x tending to 0 plus fx which can be further written as limit of x tending to 0 plus x which is equal to 0. Similarly we can find left hand side limit of the function at x is equal to 0. This is limit of x tending to 0 minus fx which can be written as limit of x tending to 0 minus x which is further equal to 0. Now clearly we can see 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 0. So we can write 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. Now using key idea we get function f is continuous at x is equal to 0. Since these two limits coincide each other and they are equal to value of the function at x is equal to 0. So the function f is continuous at x is equal to 0. Now let us take the continuity of the function at x is equal to 1. We know for x greater than 1 fx is equal to 5 and for x less than equal to 1 fx is equal to x. Now first of all let us find out the value of the function at x is equal to 1. This is equal to 1. Now we will find out right hand side limit of the function at x is equal to 1. Now this is equal to limit of x tending to 1 plus 5 which is equal to 5. We know fx is equal to 5 for x greater than 1. So we have written here 5 for fx. So we get this limit equal to 5. Now let us find out left hand side limit of the function at x is equal to 1 that is limit of x tending to 1 minus fx which is equal to limit of x tending to 1 minus x. We know for x less than 1 fx is equal to x. Now this limit is equal to 1. Clearly we can see right hand side limit is not equal to left hand side limit or we can say that two limits do not coincide each other. So the given function is not continuous at x is equal to 1. Let us now check continuity of the function at x is equal to 2. We know fx is equal to 5 for x greater than 1. So first of all let us find out value of the function at x is equal to 2 that is f2 is equal to 5. Now let us find out left hand side limit and right hand side limit of the function at x is equal to 2. Now right hand side limit is given by limit of x tending to 2 plus fx which is equal to limit of x tending to 2 plus 5 which is equal to 5 only. Now similarly we can find left hand side limit of the function at x is equal to 2 that is limit of x tending to 2 minus fx is equal to limit of x tending to 2 minus 5 which is equal to 5 only. Clearly we can see that two limits coincide each other and their value is equal to value of the function. So we can write limit of x tending to 2 plus fx is equal to limit of x tending to 2 minus fx is equal to f2 is equal to 5. Now this implies function f is continuous at x is equal to 2. So our required answer is function f is continuous at x is equal to 0 and x is equal to 2 and function f is not continuous at x is equal to 1. This completes the session hope you understood the solution take care and have a nice day.