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Problem 68 Hard Difficulty

For what values of $ x $ is $ g $ continuous?
$$ g(x) = \left\{
\begin{array}{ll}
0 & \mbox{if $ x $ is rational}\\
x & \mbox{if $ x $ is irrational}
\end{array} \right.$$

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Video Transcript

this problem Number sixty eight of the Stuart Calculus, eighth edition, section two point five or what values of X is G continuous and G of exes to pine is zero if x is rational and x X is irrational. And we recall the definition of continuity as the limit as X approaches any value, eh? Have a function, Miss Case G. Jeez, continuous. If and only if this image this limit is equal to the function g evaluated at this value, eh? And our proof will be heavily based on the behavior of G um, with respect to its limit. Two we're in a He's a quick sketch to help us understand more of what of how this function behaves on what we can look at. It is first, um, plot. The only point that is that are definitive. For example, the function g zero if X is rational. So at X equals zero, for example, that's a rational members. So the function is equal to zero at one. It's equal zero to its equal to their own. Make it a one and negative too are also rational numbers. Even halfway between each of these points will be zero because we can stay negative. Zero point five is rational. Negative One point time is rational, so a lot of the function will be zero. What we should recall is that for every rational number between every two rational numbers, there are infinitely many irrational numbers on DH. That's Ah given, Ah, that's given information where the irrational parts of the function might look. Ah, similar to the graph of G of X equals X, for example, a linear function. So let's look at it. The area around X equals one around X equals one or exactly at X equals one. We know that X equals one isn't is a rational number. That's why that functions equals zero. But we know that in the immediate vicinity of one there are irrational numbers. At one point zero one three five nine seven. You know any irrational number that you can think of that is non terminating decimal so very near to one. And if it is a rational than the function takes on that same X value. So, for example, that function would take on the value very cost one around two there are very ah, there are irrational numbers. Very very close to two. And so the function will take on its value very, very close to two as one s o. For every irrational number, we will have many points along this line of G of X, equal to X, where those irrational numbers would plot. This is articular function and the purpose of this of understanding This is to show that the function has many discontinuities because there are many variations of rational to irrational numbers in consecutive numbers. Because between any two rational numbers, there are an infinite amount of irrational numbers and between any two irrational numbers Aaron, infinite many amount of rational numbers. So the sketches only supposed to show a bit of discontinuity where it ferries between this function of X of G f X equal to X and X equals zero. So we can see that for all, for all values of X far away from this origin, there are massive discontinuities, Clearly the limit as X approaches. Any of these numbers far away from the origin of dysfunction do not equal that function value. However, we should take care of what happens exactly at the origin, because as we approach thie origin as we get closer and closer to zero, the function g of X equals X and zero Coincide. So if we imagine a very small, irrational number um, very, very close to zero in the immediate vicinity of zero, it's limit will actually be very, very close to the function evaluated at that point. So this will be true for only one point exactly at the origin of equal zero. And we confirmed that this definition of continuity the limit is exported. Eh of G of X is equal to Gaea on ly sad that this is only satisfied for the value equal zero because of the behavior of the two different functions eso this definition continuity on ly stands for a zero. But as we can see it for the rest, right, The values of X G is discontinuous