 First question, okay I have a function here, pi into gif of x minus pi by 1 plus gif, discuss the differentiability of this function, discuss things I am opening it, the entire real number, tell me the points where the function will be non-differentiable, gif of x is the greatest integer less than equal to that, this is gif of 3.5, that is 3, gif of 10.7, gif of minus 4.2 is minus 1, yes, you know this question is worth 5 seconds, so between minus 1 it is just 5 seconds, this question is just worth 5 seconds, minus 1 is completely undefined. This is not tanics actually, it is complicated, 5 seconds, tick tick 1, tick tick 2, huh, true, everything, how many of you are scared of mod function in gif function, what they will be scared about, mod function is something which makes that particular function give a positive output or they will be scared about it, you are not thinking hard enough, gif the name itself says greatest integer it will return which is less than that number, so do you think that this number would always be an integer or not, an integral what is differentiable, every 1 by mod x, see the word is the rule of life, what are the mathematics, so anything, this is not equal to 0 and 0 is option 8, this function is continuous, it is continuous for all the differentiable and it is equal to 0, please do not plot it because you know in normal exams you will not be given that phone in your hand, it is differentiable but it is not continuous and it is equal to 0, is it this one, weird option, how can be differentiable we can cross it, think properly, which is only point of contention here, 0, just check and it is 0, what is right, it will not be differentiable, so you say option p, get the 1 by 1, and it is the 1 by 1, it is the 1 by 1, it is the 1 by 1, it is the 1 by 1, it is the 1 by 1, it is the 1 by 1, it is the 1 by 1, it is the 1 by 1, it is not differentiable, it is not x is equal to 0, this is 0 and when x is greater than 0, it will be x e to the power minus 2 minus 2 by x, okay, left hand limit, what is the 0 plus x e to the power minus 2 by x, 0. What is this, disturb you, x is as it is tending to 0, it is not 0 exactly, it is a quantity which is very very small and positive, so this quantity will be almost tending to 0, his reasoning is this quantity will be smaller, very small quantity, why because it is e to the power negative infinity which is actually 1 by e to the power infinity, correct, but this quantity lesser than 1, so if you raise this, this will start going towards 0, this is also going towards 0, so both of them are becoming very small and hence the answer is 0, okay, then it will lead to another indeterminate form, that is actually infinity into 0 form, that would require a separate treatment, but here is absolutely correct that the answer is going to be 0, so left hand limit, right hand limit and the value of the function at 0, all are 0, hence it is continuous at 0, so that is the bad news for me because I have to move the differentiability test there, correct, I have not been discontinued from 0, our work would have gone over, right, but unfortunately now we have to do the differentiability test at 0, so how will I do that, simple, differentiate the function left to 0, differentiate the function right to 0, put x as 0, left to 0 if you differentiate you are going to get 1, so I can say L f dash 0 is what, 1, okay, no need to use first principle on this, second one, R f dash 0, now if we differentiate it we have to use product tool, so I can say e to the power negative 2 by x plus x e to the power negative 2 by x into 2 by x square, right, now because of this x which is remaining over here and x tends to 0, plus you can say, this entire thing will become, what will happen to this entire thing, it will be quite a big number, sure, undefined, yes, right, so in this case it will become, correct, so in this case we can say it is not a different g m L x equal to 0, but it was continuous everywhere, because once you cross that critical point it is all made up of continuous functions, so it has to be continuous everywhere, but not different shape L at x equal to 0, has anybody tried plotting this, yeah, all the long moves like this, yes, how do you do, how is it like that pages, yeah, see here is on the graph, yes, what are they looking for, what are they looking for,