 Hi and welcome to the session. Let us discuss the following question. Question says, for what value of lambda is the function defined by fx is equal to lambda multiplied by x square minus 2x? If x is less than equal to 0, fx is equal to 4x plus 1. If x is greater than 0, continuous at x is equal to 0. What about continuity at x equal to 1? First of all, let us understand that if function f is continuous at x equal to a, then limit of the function is equal to value of the function at x equal to a. This is the key idea to solve the given question. Let us now start the solution. We are given fx equal to lambda multiplied by x square minus 2x. If x is less than equal to 0, fx is equal to 4x plus 1. If x is greater than 0, we are given function is continuous at x equal to 0. So, let us find out right hand side limit of the function at x 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 4x plus 1. Now, this is equal to 1. Let us now find out left hand side limit of the function that is limit of x tending to 0 minus fx which can be written as limit of x tending to 0 minus lambda multiplied by x square minus 2x. Now, this is equal to lambda multiplied by 0 square minus 2 multiplied by 0. This is equal to lambda multiplied by 0. Now, we know function is continuous at x is equal to 0. Only is right hand side limit is equal to left hand side limit of the function at x 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. Now, substituting the corresponding values of the two limits, we get 1 is equal to lambda multiplied by 0 or we can say 1 is equal to 0 which is not possible. Hence, we get for no value of lambda function is continuous at x equal to 0. Let us now discuss the continuity of the function at x equal to 1. Now, we know at x equal to 1 given function is defined. Let us now find out right hand side limit of the function at x equal to 1. So, we get limit of x tending to 1 plus fx is equal to limit of x tending to 1 plus 4x plus 1 which is equal to 4 multiplied by 1 plus 1 which is further equal to 5. So, we get limit of x tending to 1 plus fx is equal to 5. Let us now find out left hand side limit of the function at x equal to 1. So, we can write limit of x tending to 1 minus fx is equal to limit of x tending to 1 minus 4x plus 1. Now, this is again equal to 5. So, we get limit of x tending to 1 minus fx is equal to 5. Let us now find out f1 f1 is equal to 4 multiplied by 1 plus 1 which is equal to 5. Clearly, we can see right hand side limit is equal to left hand side limit is equal to value of the function at x equal to 1. This implies given function f is continuous at x equal to 1 for any value of lambda. So, our required answer is for no value of lambda function f is continuous at x equal to 0, but function f is continuous at x equal to 1 for any value of lambda. This completes the session. Hope you understood the session. Take care and goodbye.