 Next question. F of the IEF function. The reason why I am giving you so many IEF functions is because I want to discuss the differentiability of this function. Discuss the differentiability of this function. Right? Now you know in exam when it comes in comedy exams 90% Janta what do you do? JIA function, JIA function, non-differentiable, non-differentiable, obviously non-differentiable. But we learnt two non-differentiable, can't say, depends upon situation. Do you have options? This function f of x, this function f of x, non-differentiable or you can say, let's say I take, let me know to give you options because then it will not make it that challenging. Okay, just do the differentiability of this function. Discuss the differentiability of this function. In fact, I think the question is slightly more difficult. Discuss the continuity and differentiability of this function. Discuss the continuity because anyway in general. So, telling the points where you feel this function is discontinuous. And integral points it will be discontinuous. Telling the points where this function will be discontinuous. Whenever JIA comes you start doubting integer points, right? Discontinuous and integer points. Or integer points. Okay, so let's say I do two, okay. Let's check the left, let's check the right hand limit at two. Let's check the value of the function at two, okay. What is the left hand limit at two? Left hand limit at two means when you are slightly 1.9, right? 1.9 JIA is what? 1, 1 square is 1 minus. 1.9 square is what? Definitely less than 4, correct? What are the JIAs of that? 3. So, what's your answer? Negative. Let's check the right hand limit at two. Take the numbers 2. What is the answer? Zero. I don't have to check what is the value of 2 also. That means it X equal to 2. Does this mean it will be discontinuous at all integer points? It was just for two, right? Yes. Many of us would like to see the points. What is the test at one? Test at one, absolutely. Now, you will be surprised when you test it at one. How do we get smaller? Let's test it at one. At one, you'll see you will be surprised. Let's find the left hand limit at one. Okay. Slightly less than one, what will be JIA of X? Zero. Zero square will be? Zero. Minus. 0.9 square, 0.81. That will be zero. What will be the right hand limit? Slightly more than one. One. Slightly more than one square. And value at one? Yes. So, we cannot say blindly that for all integer points it is discontinuous. It is actually continuous at one. So, how would I state this? The function is discontinuous at all integer points except one. Except one. Zero and only checked, it will be discontinuous. So, now, points it will be non-differentiable also. That means I have to check its sensitivity at one. That means left hand derivative and right hand derivative at one. Yes sir. So, how would I define this function? Slightly left to one and slightly right to one. Can we fall back to our redefining the function or you want to do your first principles? The call is yours. So, we need to find out the differentiability of this function. This was the function, right? Yes, the differentiability at X equal to one. I can redefine this function. Very easy. When X is slightly less than one, when X is slightly greater than one, now you will only tell me. When X is slightly less than one, this is the function. When X is slightly less than one, zero. What is the outcome of zero? Zero. Zero. Slightly less than one. Yes, zero will be zero. It will be zero at both the places, but I am born in terms of X. Can you write it in terms of X? Zero. X minus one. Yes, slightly less than one, you said zero, right? Slightly higher than zero. Correct? That is the constant function? Yes. So, it will be differentiable also? No, no, no. I want a completely flat. I want a completely flat. Show me your mistake. I want a flat. Yes, see at one, it is completely flat. You can always draw a unique tangent with a finite scope. Correct? Okay. So, it has to be differentiable at one. Yes, it is differentiable at one. Is that fine? Yes. It is continuous and indifferentiable. Next question. Zero. X is equal to what? It is a flat, no? It is a flat line. X is equal to what? A flat line, if you draw a tangent, there will be also a flat line. The region can be expressed in terms of X, but G is here till now. So, you have to write some integer in place of this. Right? Slightly different type of question. But that is not difficult. Mine function, those words, max function. In maths, how many of you have taken computations? Sir, what? Okay. How many of you have taken computations? I know. Oh, yeah? I know. Exactly. Okay. Anyway, this is a function which actually returns you for particular X, the maximum of these three values. Understood? Let's say I put X, this returns to me 5, this returns to me 3. This will return to me. Which is the max of this? This. That's what the question says. So, for a given X, whatever X you want to put in, it will return the max of these three values. Highest overlaping. I know. I know overlaping. One of them will give you maximum, that will be the, that will be the. Okay. Are you getting it? Yes. Return the max of the values. Correct? So, you put in the argument. It will check which of them is giving you max. It will return that as your answer. Okay. I'm talking in terms of C plus plus. The computations that you write. In terms of C plus plus. My question here is, discuss the points where the function is not differentiated. Or find the points where the function is not differentiated. Find the points where f of x is not differentiated. It's a zero. Now, the moment friends look at it, they get confused. How do I know this function actually? How do I grab this function? Do you take all k? That's the challenge. How do you do it? It's a switch case. You know your differentiability criteria, but the function itself will be difficult for you to apply your differentiability criteria. That's the reason. That's any complication. There are many such points. I am not sure. For the few who agrees, they are telling the method to do it. Sir, wait. Shall we do it? This is your maths. Imagine they are doing a random function. Random function means they will give you maths. I'll tell you for both the cases. First of all, everybody. No doubt about that. Again, I'll repeat this. So, this function will take some input. Then we'll emphasize the answer. This is a function which is a special kind of function. Now, when dealing with this function or its cousin, which is the middle of this, we follow the graphical approach only. Where we plot all... So, mod x plus... Okay. Mod x minus 2. Can we shift it? When it was x plus 2, mod x plus 2 to the left. In this case, it will be to the right. That will this point be? This point be? Okay. How do I plot 2 minus mod x minus 1? Now, again, for people who have not done the transformation of graphs with us last year, please follow this now. When you say, I want to draw this graph, first draw those basic graphs that you can make for it. This graph has evolved from this graph. Correct? So, it's not a composition. It's not a composition. It's the kind of a transformation that you apply to that graph. So, first draw the very basic graph that you can draw for this. Which is this? Yes or no? Do you think? I like it. Okay. This graph is like this. Yes or no? Correct? What will happen if I put a minus sign in front of it? It will become a graph. It will be a reflection and x. It will reflect about the x axis? Yes. Put it out. Now, if I make x, if I change my x with x minus 1, what will happen to this graph? It will shift one into the right. Absolutely. It will shift one into the right. Can I draw it like this? Correct? When I add a mover here, it's what I'm drawing. When I'm giving a push, it will take a jump off to above. So, can I say the graph will now look like this? Then add it. So, if I have to produce this over here, the graph would have failed like this. How will they? Very good question. See, this point will be x equal to 1, right? Right? 1 plus 1. But at x equal to 1, it has to be x equal to 1. That's right. Now, I've told you about all the three graphs in one single x-y coordinate system. Now, what is the max? Max means, max means, just trace that part of the graph which is the top graph. What will be that? Can I say it will come like, this is the boss so far? The moment it reaches, this becomes the boss. The moment it reaches? Sir, it's not differentiable. Is that fine? So, you're fine? Are you getting this point? Yeah. Now, tell me where or it is non-differentiable. See, the point is 0.5 and 0.9. Can I say this point is not differentiable? No. You want to suffice to know it's half? Yeah. Now, many people say, sir, how do you figure out when the graph is half? Symmetry of the figure is fine. But let's say the person who is not able to understand that how does it come out half? It's actually the meeting point of this line and this line. Correct. Now, by the way, what is this line? y is equal to 2 minus x. What is this line? This is y is equal to x plus 1. Solve them simultaneously. That is 2 minus x is x plus 1. So, 2x is equal to 1. So, x is equal to? That's how you get that one. It's not 1 also. So, half is one of your answer. One is one of your answer. Is it not differentiable at this point? Yeah. What's that point? 3 by 2. Yeah. 3 by 2. 3 by 2. Sure? Yeah. People ask you, test the differentiability now of... Yeah. The first point is... Which color do you want? No. Red. Red or pink? Doesn't matter. Doesn't matter. It doesn't matter. Min will be this. Correct? Yeah. Can I say it is non-differentiable at minus half? Yeah. Zero. This is three. So, this is two and a half. Is that right? Yeah. Don't be scared of these functions. They are very easy. Start with this. You can't start without them. Very difficult. Because, you have to see which is exceeding the other. You have to see a lot of permutation combinations that will be better. Take a break. On the other side of the break, we'll take a few more questions and start with the inverse trick.