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