 Hi and welcome to the session. My name is Shashi and I am going to help you with the following question. Question says, find all points of discontinuity of f where f is defined by fx is equal to x upon modulus of x if x is less than 0, fx is equal to minus 1 if x is greater than equal to 0. First of all let us understand that function f is discontinuous at x is equal to a if limit of x tending to a plus fx is not equal to limit of x tending to a minus fx. This is the key idea to solve the given question clearly we can see the function is discontinuous at x is equal to a only if the left hand side limit is not equal to right hand side limit of the function at x is equal to a. Let us now start the solution we are given fx is equal to x upon modulus of x if x is less than 0 and fx is equal to minus 1 if x is greater than equal to 0 we are given fx is equal to x upon modulus of x if x is less than 0 x is a polynomial function so it is continuous at every real number modulus of x is modulus function so it is also continuous at every real number this implies function f is continuous at every real number less than 0. Now we are given fx is equal to minus 1 for x greater than 0 now 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 0 now let us check if the function is continuous at x is equal to 0 now clearly we can see the function is defined at x is equal to 0 so we can write at x is equal to 0 function is defined now let us find the 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 minus 1 which is equal to minus 1 now let us find out left hand side limit that is limit of x tending to 0 minus fx is equal to limit of x tending to 0 minus x upon modulus of x now this is equal to limit of x tending to 0 minus x upon minus x we know for x less than 0 modulus of x is equal to minus x now this is equal to limit of x tending to 0 minus minus 1 this is further equal to minus 1 so we get limit of x tending to 0 minus fx is equal to minus 1 now clearly we can see right hand side limit is equal to left hand side limit let us now find value of the function at x is equal to 0 so we can write f0 is equal to minus 1 so we get 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 minus 1 so this implies function f is continuous at x is equal to 0 we have already shown that function f is continuous at all the real numbers greater than 0 less than 0 and equal to 0 so this implies there is no point of discontinuity so our required answer is function f has no point of discontinuity this is our required answer this completes the session hope you understood the session goodbye