00:01
So we have the function f of x equals the absolute value of x minus 3.
00:05
And we wanted to determine whether this function is continuous at x equals 3.
00:13
So absolute value functions can always be written as piecewise functions.
00:19
Because remember, when x minus 3 gives a negative number, then this can be written as negative x minus 3.
00:31
3.
00:32
Because when x minus 3 is negative, then the opposite of that is positive and absolute value always gives out positive numbers.
00:40
When x minus 3 gives out a positive number, then the absolute value of that is just going to be the same as x minus 3.
00:49
So this function f of x can be written as a piecewise function where f of x equals negative x minus 3 whenever x minus 3 is negative, which is whenever x is less than 3, or f of x equals x minus 3, where x is greater than or equal to 3.
01:23
So now that we've written it as a piecewise function, we can now determine continuity fairly simply by showing three things.
01:32
1, f of 3 exists, 2, the limit of f of x as x approaches 3 exists, and they're equal to one another.
02:00
So let's show those three things.
02:01
So first of all, f of 3, we know is just going to equal 0, because even if we look at the original function we had, the absolute value of 3 minus 3 is just 0, so we can leave it at that.
02:14
So great, f3 exists...