Of I, it is an exercise to show that f is continuous at a if...

of $I$, it is an exercise to show that $f$ is continuous at $a$ if and only if $f(a) = \inf \{f(x) : x \in I, a < x\}$ or if and only if $f(a) = \lim_{x \to a^+} f$. Similar conditions apply at a right endpoint, and for decreasing functions.

Transcript

00:02 Let's now prove that the absolute value function, so iphobax equal absolute value of x, let's prove that it's continuous everywhere.

00:29 So let's see.

00:37 Well, we know that f of x is a piecewise function, the absolute value can be rewritten as x, the absolute value of x is just equal x if x is greater equal 0 is equal minus x if x is less than 0.

01:05 So both of these two functions, x are linear functions, x and minus x.

01:14 So they are continuous everywhere.

01:17 The only point of concern might be x equals zero, but it's clear that limit when x approaches zero of f of x, well, it's zero either from the right or from the left.

01:41 It's just zero all the time, and this is equal to the value.

01:46 Of the function at x equals 0 is equal to f of 0, absolute value of 0.

01:56 So just by using the definition of continuity, you can easily prove that the absolute value function, even if it's a piecewise function, but it's any way continuous everywhere, zero included.

02:14 Now, let's try to prove.

02:17 In a more general way that if we know that a general function f of x is continuous on a given interval, then we can also say that the absolute value of f of x is a continuous function.

02:48 Well, i think it's quite straightforward because as we saw here, at the beginning, i mean, we can always rewrite the absolute value function as a piecewise function.

03:07 So in some way, the absolute value function, it's a composition of two functions.

03:16 It's a composite function, where we do combine the f function and the absolute value function and the absolute value function.

03:27 So it's just a composition of two functions and they are just both continuous function.

03:38 The absolute value function we just show it's a continuous function.

03:43 And if we already know that f of x is continuous, well we just have to remember that if f and g are two continuous function, then the composition of the two functions, f and g, which means f of g of x, is still a continuous function.

04:10 So in this case, thinking about the absolute value or any function f of x, well, if we call the absolute value function, let's say we call it capital f and we have just our function f of x, then in this case we can say that f of capital f of x is just a continuous function because it's a composite function and it's made of two continuous functions function, small half of x, which we know is continuous, and absolute value function, which will just prove is a continuous function.

05:09 So just the composition of two continuous function.

05:13 Now let's see if it, we can also say, well, let's clear a little bit, if we can say also, also, you know, the except the opposite...

Question

of $I$, it is an exercise to show that $f$ is continuous at $a$ if and only if $f(a) = \inf \{f(x) : x \in I, a < x\}$ or if and only if $f(a) = \lim_{x \to a^+} f$. Similar conditions apply at a right endpoint, and for decreasing functions.

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