Confirming continuity over an interval
Functions continuous at specific x-values
- [Voiceover] Which of the following functions are continuous at x equals three? Well, as we said in the previous video, in the previous example, in order to be continuous at a point, you at least have to be defined at that point. We saw our definition of continuity, f is continuous at a, if and only if, the limit of f as x approaches a is equal to f of a. So, over here, in this case, we could say that a function is continuous at x equals three, so f is continuous at x equals three, if and only if the limit as x approaches three of f of x, is equal to f of three. Now let's look at this first function right over here. Natural log of x minus three. Well, try to evaluate it, and it's not an f now, it's g, try to evaluate g of three. G of three, let me write it here, g of three is equal to the natural log of zero. Three minus three. This is not defined. You can't raise e to any power to get to zero. You can try to go to, you could say, negative infinity, but that's not, this is not defined. And so, if this isn't even defined at x equals three, there's no way that it's going to be continuous at x equals three, so we could rule this one out. Now f of x is equal to e to the x minus three. Well this is just a shifted over version of e to the x. This is defined for all real numbers, and as we saw in the previous example, it's reasonable to say it's continuous for all real numbers, and you could even do this little test here. The limit of e to the x minus three as x approaches three, well that is going to be, that is going to be e to the three minus three, or e to the zero, or one. And so f is the only one that is continuous. And once again, it's good to think about what's going on here visually, if you like. Both of these are, you could think of them, this is a shifted over version of ln of x, this is a shifted over version of e to the x, and so if we like, we could draw ourselves some axes, so that's our y-axis, this is our x-axis, and actually, let me draw some points here. So that's one, that is one, that is two, three, two, and three, and let's see, I said these are shifted over versions, so actually, this is maybe not the best way to draw it, so let me draw it, this is one, two, three, four, five, and six. And on this axis, I won't make 'em on the same scale, let's say this is one, two, three. I'm gonna draw one, two, three, I'm gonna draw a dotted line right over here. So g of x, ln of x minus three is gonna look something like this. If you put three in it, it's not defined, if you put four in it, ln of four, well, that's gonna, sorry, ln of four minus one, so that's gonna be ln of four minus three, is actually let me just draw a table here, I know I'm confusing you. So, if I say x and I say g of x, so at three, you're undefined. At four, this is ln of one, ln of one, which is equal to zero, so it's right over there. So g of x is gonna look something like, something like that. And so you can see at three, you have this discontinuity there, it's not even defined to the left of three. Now f of x is a little bit more straightforward. If you have, so e to the three is going to be, sorry, f of three is going to be e to the three minus three, or e to the zero, so it's going to be one, so it's gonna look something like this, it's gonna look something, something like, like that. There's no jumps, there's no gaps, it is going to be continuous, and frankly, all real numbers so for sure it's going to be continuous at three.
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