 Hello and welcome to the session, let's work out the following question. It says proof that the function given by fx is equal to mod of x minus 1 where x belongs to the set of real numbers is not differentiable at x is equal to 1. So let's now move on to the solution. The given function is mod of x minus 1 function can be written as minus of x minus 1 if x is less than 1 and it is x minus 1 if x is greater than 1. This is by the definition of mod x which says mod x is equal to minus x if x is less than 0 and it is x if x is greater than 0. Now we have to show that it is not differentiable at x is equal to 1. Now we know that the function f is differentiable is equal to c if limit h approaching from the left hand side of f of c plus h minus f of c upon h is equal to limit h approaching from the right hand side of f of c plus h minus fc upon h. If these two limits are equal then we say that the function is differentiable at x is equal to c. Now we will check the differentiability at x is equal to 1. Now we will first find the left hand limit h approaching from the left hand side of f of 1 plus h that is c plus h here c is 1 minus fc that is f of 1 upon h. Now since h is approaching from the left hand side when x is less than 1 the function will be this. So we have limit h approaching from the left hand side f of 1 plus h. So we will put 1 plus h in place of x here. So we have minus of 1 plus h minus 1 minus f of 1 we will put 1 here. So it is minus of minus 1 minus 1 upon h this is minus h and this is 0. So it is minus h upon h. So the limit is minus 1. Now we will find the left hand limit h approaching from the right hand side of f of 1 plus h minus f of 1 upon h. Now since h is approaching from the right hand side when x is greater than 1 so function will be this. So we have limit h approaching from the right hand side f of 1 plus h minus h. So we put 1 plus h in place of x. So we have 1 plus h minus 1 minus f of 1 that is 1 minus 1 upon h this is limit h approaching from the right hand side of h upon h h gets cancelled with h and the limit would be 1. Therefore limit h approaching from the left hand side of f of 1 plus h minus f of 1 upon h is not equal to the limit h approaching from the left hand side this is right hand limit and this is left hand limit. So the left hand limit is not equal to the right hand limit. Therefore function f x is not differentiable x is equal to 1. So this completes the question of this session. Bye for now. Take care.