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Problem 72 Hard Difficulty

(a) Show that the absolute value function $ F(x) = | x | $ is continuous everywhere.
(b) Prove that if $ f $ is a continuous function on an interval, then so is $ | f | $.
(c) Is the converse of the statement in part (b) also true? In other words, if $ | f | $ is continuous, does it follow that $ f $ is continuous? If so, prove it. If not, find a counterexample.

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Video Transcript

the first problems we want to prove the absolute value function is continuous everywhere. So first, so this is the absence function. And to say we want to prove um f of X is continuous at one point say execute A. Then we can see the difference of the function. So this if you write it is actually echo to absolute value absolute value of x minus absolute value of A. And if you use a triangle inequality you can prove this is smaller, then absolutely a value of x minus A. So when X 10 to a. You'll see X -1-0. So this means because this function is the other bands with sometimes goes to Zeros and F of the x minus F. A. So this one also tends to zero. So that means the function F is continuous at two executive A. And his hair is like an arbitrary so ffx is continues everywhere. And for the second province. So we weren't proves that fs continues then similarly for absolute value of X. So this is also easy to prove, say so if we want to through F of X is continuously at X equal to a. So what we do is so the absolute value of F at X man is absolute value of a. The difference of the function difference. This is also smarter by the triangle inequality is smarter than ffx minus F. A. Since we know if is a it continues, that means This quantity 10- zero when x. 10 to a. And this tells you this time goes to there, that means fx is continuous. So for the last province that we want to know, the answer is not true. So it is forced. And we want to find a counter examples into the examples. Also you notify. So one thing you can think uh say fo Becks If you could to one. The X is The rational numbers and -1. When X is like a Nazi rational numbers. So this function clearly it's not continues because you can just and draw the figure. So there are some dots here. So this is one and also minus one and some thoughts here. This function is not continuous. Lecture once you take the absolute value of FX. So this is Echo to save one. Right? So this is a customer of functions so it is continuous. So this means we have a country example. So after value is continues and here is a constant that the the F of X is not continues. So the answer is forced.