Okay. What we want, Thio, what we're starting with is, um f of X is equal to one. If x is rational and zero if X is irrational, okay. And so we want to show that f of X is discontinuous at every point. Okay? And so if we want to show continuity the definition Ah, continuity, um says that, um f of sea. So your function evaluated at some X value, um, has to be defined. And the limit as X approaches see of your function has to exist And those to the function evaluated at that see value, Hasi, with the limit as X approaches, see of the function. So that is the definition of continuity. And so what we need to do is to show that if the functions discontinuous at every point that it does, this definition of continuity does not hold true. And so let's go ahead and get started. Um, we're going to use, um the fact that non empty um interval, um, of real numbers, um, contain, um, both rational and irrational numbers. Okay. And so we're gonna let, um, we're gonna let, um, first of all, if we're gonna let see, um B um, a rational number. And I'm gonna do it for the rational number case. But it can be still shown We can do it again for irrational numbers is gonna hold true. And so, um, if I have my function evaluated at some rational, um, number, then that is going to equal wine. Now, if I let the limit as X supposed approaches, see of that rational number, this does not exist. And it is because when we're approaching this rational number, we are having to, um, go through an interval that is comprised of rational and e rational numbers. And so, when I am approaching the sea value, um, I can go from a function value of 101010 so it will never settle on any value. So this limit does not exist. And so therefore, because those two do not equal each other and more importantly, this too thought exist, then my, um, function is discontinuous. Um, for that see value or more importantly, for every, um, point. And so we can also kind of show this for, um, you Russell numbers as well. Now, another question poses itself because really, what? We're doing here? We were looking at, um, ex approaching. See, we're really in our mind's eye looking at it as we're approaching it from the left and from the right. And so if I'm approaching it from the left, I could be approaching. When I get arbitrarily close to see, I could be landing on an irrational number, which means I'm approaching zero. Whereas if I'm looking at it from the right, I could be arbitrarily approaching a rational number, which then my wife values approaching one, so I will never get connected at the same value. And so this can be, um this can be, um Also look at, um the question is is f of X, um, right, continuous or left continuous. And the answer is no by the same justification. Um, because if, um, the limit as X approaches see, from the right of the function as well does not exist because we are arbitrarily approaching C and so see bounces ASM approaching. See from the right, I am. I am bouncing from rational to irrational numbers so it will never approach a single value. And the same can be true for the limit as X approaches see from the left as well does not exist, and since those right and left limits do not exist, then the function is not continuous, either right, continuous or left continuous.