hello and welcome back to ordinary differential equations the video series where we talk about solutions for such equations given by derivatives and in today's Part 9 we will talk about the uniqueness of solutions of a given initial value problem and the key ingredient for that will be the so-called Lipschitz continuity more precisely here we will talk about locally Lipschitz continuous functions however you already know before we start first I want to thank all the nice supporters on study here on YouTube or on patreon you make it possible that I can create such maths videos here okay then let's start with the topic of today which is about lip shit's continuous functions and in order to get the idea let's start with an ordinary function from R into r and now you should already know from real analysis that we have two important properties for such a function namely we have the continuity on the one hand and the differentiability on the other hand or even a stronger property would be that f is continuously differentiable but either way we know that differentiability implies continuity and now I can already tell you that the Lipschitz continuity is the middle Crown between these two Notions here so F being lip shit's continuous is more than just F being continuous but still less than saying that f is continuously differentiable or more concretely the notion we will now Define is saying that f is a locally Lipschitz continuous function therefore at this point you should immediately remember that for these three definitions for these three properties we have these implications and now it turns out that this new Middle Ground is exactly what we need what we want to solve ordinary differential equations in a unique way hence I would say let's go to the definition of this Lipschitz continuity okay so here we have the formal definition now and let's formulate it with the notations we already used in Odes this means we consider a map V from RN into RN again and you already know sometimes the domain of definition is smaller than the whole RN and then we just choose an open set u however that does not change the definition at all so we keep it simple here if the domain chosen srn and now the function V is called locally ellipse it's continuous if it fulfills a local condition at each point in the domain hence we would write the quantifier for all X in the domain something has to be fulfilled and you see I want to use a compact notation so I use the quantifier and the variable below and as always in such formulations we have the for all quantifier and there exists quantifier okay and now you might already know a local condition means that we find that there exists a local neighborhood of this point x and since we live in RN we can always choose an Epsilon ball hence for each point x we find an Epsilon greater than 0 such that the whole Epsilon ball around X still lies in the domain however the domain was not a crucial part here so let's immediately go to the point we actually want to discuss namely we want to discuss the difference between two values of V so let's write V of Y minus V of Z and this difference we just measure in the standard Norm of RN and now Lipschitz continuity always means that we have an estimate for this distance more precisely you should see here on the left hand side we have a distance for the outputs and now we want an estimate with respect to the distance of the inputs in other words on the right we now have y minus Z and of course also measured with respect to the standard Norm in RN however now the concept of Lipschitz continuity allows that there is a constant L included moreover The crucial Point here is that this capital L works for all Y and Z simultaneously so in other words you write there exists such an L greater than Zebo such that for all Y and Z the following property holds and now the thing that makes this local is that we only need that inside our Epsilon ball therefore you can simply remember we find an Epsilon ball and a constant L such that we have this estimate for all Y and Z inside the Epsilon ball and then usually this L we can simply call the Lipschitz constant okay with that now you know what it means if we call a function V locally ellipse it's continuous and now from this definition we can immediately conclude two important things you definitely should remember indeed we have already discussed that in the beginning of the video but now we can actually prove this fact first if you have a locally lipsticks continuous function it's definitely also an ordinary continuous function in fact this is easy to prove just assume you have a sequence of inputs and maybe we call them y n and now if this sequence y n converges to another point Y we want to show that the images also converge there please recall this is exactly the definition of continuity however here we already know we can measure the distance of the images by using the distance of the inputs and now since the right hand side here goes to 0 when n goes to Infinity also the left hand side has to go to zero in other words we have the convergence of the images as well and this immediately proves the continuity at each point in the domain okay then in the next step let's talk about the connection to the differentiability of a function again let's assume we have a locally Ellipsis continuous function which means we can use this estimate here and now you can say it's not a problem at all let's bring the distance of the input space to the left hand side as well of course for distinct points Y and Z this is always possible hence what we see is that this difference quotient here on the left hand side is now less or equal than our constant l however please don't forget the constant L is the same for all points Y and Z in the neighborhood hence we conclude that the locus slopes we can calculate with this difference quotient here are all bounded by the constant l so this means locally the slopes of the function cannot go to Infinity indeed this is a very important property of the Ellipsis continuity we will use in our ode course here however before we apply these Notions to differential equations let's first look at differentiable functions more precisely I want to take a C1 function so I continuously differentiable function and moreover to keep it simple let's use the one-dimensional case here in fact that is also the reason I call the function f instead of V again okay now let's fix the point x and an Epsilon neighborhood around it and then we know we can just look at the secant slope from before however now in the one-dimensional case which means the norm is simply the absolute value and now you might already recognize for a differentiable function here we can use the mean value theorem this one implies that the secant slope here on the left hand side can be written as a tangent slope in other words we can use the derivative of f the only thing we need here is an intermediate point we can call C and you know it's intermediate point so it lies between y and z so if you're not familiar with the mean value theorem you should watch my read analysis course again okay but here we have to add something because we have the absolute value on the left hand side we also need it on the right hand side and indeed this makes it better because now we can estimate the right hand side as well for example we could say we go through all possibilities this intermediate Point could have therefore let's call the new variable C tilde and let's go through all the points in our Epsilon neighborhood hence we know we will definitely hit the correct C on the way however now this supremum here which exists we can just call capital l moreover at this point please note F Prime is by assumption a continuous function and therefore the supremum here is definitely a finite number in other words this is a well-defined non-negative reader number now hence with that we have the Lipschitz constant we need for the definition moreover here I should tell you it was not written in a definition above but of course the constant could also be just zero it doesn't change anything because you can always take a bigger constant if you already have one but of course equal to 0 would be the simplest case nevertheless the conclusion here is definitely that our C1 function f is also locally lip shit's continuous therefore with that proof you already know a lot of examples for locally lipsticks continuous functions indeed this is very helpful because this is what we can use in our discussion about ordinary differential equations and here please note this whole proof here in point three also works similarly in higher dimensions and in the next video we can use that and prove that for initial value problems where the function V is locally Ellipsis continuous we have indeed a uniqueness of the solutions so I would say let's meet there and see you soon bye bye [Music]