 Hello and welcome to the session. In this session we discuss the following question that says show that the function f defined as follows is continuous at x equal to 2 but not differentiable whereas the function fx is defined as fx equal to 3x minus 2 when x is greater than 0 and less than equal to 1 2x squared minus x when x is greater than 1 and less than equal to 2 5x minus 4 when x is greater than 2 Before we move on to the solution, let's discuss the conditions when a function is continuous and differentiable First we have a function fx is continuous at x equal to a if the right hand limit given as limit x tends to a plus fx is equal to the left hand limit given as limit x tends to a minus fx is equal to f of a Now next we have a function fx is differentiable x equal to a if the right hand derivative the function fx at x equal to a given as limit h tends to 0 f of a plus h minus f of a This one upon h is equal to the left hand derivative of the function fx at x equal to a given as limit h tends to 0 f of a minus h minus f of a this one upon minus h And the common value of the right hand derivative and the left hand derivative is denoted by f dash a this is the key idea that we use for this question Let's proceed with the solution now we are given the function fx defined as 3x minus 2 when x is greater than 0 and less than equal to 1 2x square minus x when x is greater than 1 less than equal to 2 and 5x minus 4 when x is greater than 2 We have to show that this function fx is continuous at x equal to 2 but it is not differentiable at x equal to 2 first of all let's check for the continuity of the function fx at x equal to 2 So first we have to show that the left hand limit and the right hand limit of the function at x equal to 2 are equal and each is equal to f of 2 Now first we have the left hand limit of the function at x equal to 2 is given by limit x tends to 2 minus f of x now x tends to 2 minus f by sub value of x is less than 2 And for the function fx defined we find that for x less than 2 we have fx as 2x square minus x so in place of fx where we write 2x square minus x is to 2 minus x equal to therefore the left hand limit would be equal to limit h tends to 0 And in the function fx we have limit h tends to 0 2 into 2 minus h whole square minus f 2 minus f this is where the equal to 2 is to 0 2 into 4 4h minus 2 that is before the h tends to 0 8h minus 2 thus limit h tends to 0 7h tends to 0 so we put h equal to 0 in this function which gives us 2 into 0 minus 7 into 0 plus 6 which is equal to So we have the left hand limit the right hand limit the value of the function fx for x greater than 2 fx minus 4 so in place of fx we will put 5x minus 4 Right hand limit rx equal to 2 is equal to limit fx to 2 we have h tends to 0 the right hand limit is given by limit h tends to 0 and in place of x where we will put 2 into 0 10 plus 5h minus 4 which is equal to limit h tends to 0 5h plus so we put h equal to 0 in this function which is equal to 5 into 0 plus 6 which is equal to right hand limit Fine we have for x equal to 2 the function fx is which is equal to 8 minus 2 that is 6 so f of 2 is equal to the right hand limit is also 6 so equal to limit x tends to is equal to h is equal to the function fx is continuous x equal to key idea We know that the function fx is differentiable at x equal to a is the left hand derivative and the right hand derivative are equal and they are equal to the right hand derivative that is rf dash a function fx at x equal to a and this is equal to x to 0 h minus f of a this one upon h take the differentiability at x equal to 2 so in place of a 2 is equal to x to 0 f of 2 plus h minus f of 2 this one upon h fx greater than 2 we have fx as 5x minus is equal to limit h tends to 0 the function fx at x equal to 2 equal to 2 the function fx is 2x square minus x where we put the value of x as 2 to get the value of f of 2 2 the whole and this one upon 10 plus 5h minus 4 h minus 2 is 6 so we have 6 and this one upon 6 cancels so we have 5h upon h this h and h cancels so we are left with 5 that is the right hand derivative of the function fx now let's find out the left hand derivative of the function fx at x equal to 2 to is equal to limit minus f of 2 this one upon minus h for the function f to look out for the value of the function fx when x is less than 2 for x less than 2 we have 2x square minus x as the function fx equal to limit h to whole square which is equal to 6 so here we have 6 and this whole upon minus h this gives us limit h tends to 0 4 plus h square minus 4 h to the whole minus 2 plus the whole and this whole upon minus h further we get limit h tends to 0 8 plus 2h square which is equal to limit h tends to 0 2h square minus 7h and this one upon minus h which is equal to limit h tends to 0 taking h common from the numerator we have 2h minus 7 the whole upon minus h this hh cancels and so this is equal to limit h tends to 0 minus of 2h minus 7 is 7 minus 2h and the place of h will put 0 so this is equal to select 10 derivative of the function fx at x equal to 2 so the function fx at x equal to 2 is not equal to 7 that of the left hand derivative of the function fx at x equal to 2 that the right hand derivative of the function fx at x equal to 2 is not equal to the left hand derivative of the function fx at x equal to 2 therefore the function fx are differentiable a session hope you have understood the solution of the question