 is defined as x minus 1, where x is less than 0 and is defined as x square minus 2x, this is x greater than or equal to 0. I have mixed two questions for you, discuss the continuity and differentiability of f of mod x, that is first question, mod of f of x. This is the continuity and differentiability of f of mod x and mod of f of x, the best way is for the two graphs, both the graphs are easy to follow, one is the linear equation, other is a quadratic. The second question is, the second question is continuous but not differentiated. At what points? Discuss means you have to tell the points, you have to highlight the points where you are facing problems. Yes, but not the centimeter minus 1. Not the centimeter minus 1, keep writing it so that we can discuss. Thank you. Does everybody know how to plot the graph of when everything is modded? Yeah, everything is modded. Everything is modded in the graph. Everything is modded in the graph. It goes out, very good. What happens, you know? That means everything is modded on this side and whatever was originally on this side gets erased, understood? Again, I am repeating this. Let's say I give an example. Let's say this was my graph, I mean just making some arbitrary figure. This was my graph of a function. How is the graph not extra minus 1? I guess. Okay, let's say this was my function. Okay. If this is the graph of f of x, how would the graph of mod of f of x look like? It's not bad. Can we see it will look like this? Yeah. Okay. How would the graph of f of mod x look like? It is. Whatever is possible. It's not a scale graph. So many of us think graph is optional, we are also arrogant. Sometimes that's the only way of solving it. Okay, into pieces. But you can do it. Okay. Okay. This graph of f of x. We have time. Okay. Let's give you a square of x. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. First plot the graph without mod A. Minus one is just moving. Okay. This will be solid. This will be solid. Right? Yes or no? No? The first question. Discuss the continuity of x. Whatever is on the right hand side, the same will go on the left hand side. So, how does the graph look like then? It is not very nice shape but correct. By the way, it is continuous everywhere. Do you agree with that? Yeah. But not differentiable and equal to the same. Definitely not. So, use graphs wherever you can because it really makes your life very, very easy. It is worth thousand words. Now, what about perfect? So, this graph will be like this and this graph will be yes or no? Yes or no? Now, do you see? This is discontinuous. Discontinuous. Discontinuous at 0 and hence non-differentiable and at this point what happens? It is continuous but not differentiable. Is the idea clear? Discontinuous at 0. This graph, which graph is not important. So, it is continuous everywhere. No, it is not. I can plot the function. This function can be plotted for one. You have selected a point there. So, this is a graph of one. Now, we have integral points that we get smaller. What is not an integral point? See this integral point. Okay. This is for the story about mon of f of x. What about we do composition of x and make it as mon of f of mod x? Very good. Can we have that question then? What is the mon of f of x? But are you good with compositions? No, not good. This is mon of f of x. Which one? Mon of f of mod x. Mon of f of mod x will be just these two will go out. So, it will become something like this. Mon of f of mod x will become like this. Okay. Now, this also will become a point of non-differentiability, non-differentiability, non-differentiability. Understood? So, let us end this chapter now and I am sure most of you would have now appreciated what is the actual meaning of continuity and what is the actual meaning of differentiability and how we can actually use our graphs to our benefit for our advantage wherever we can and to do the differentiability test at any point. So, the best is to use a mix and match of everything to solve the problem. Is that what you are having? So, with this sequence, this chapter.