 Let's take a look at an example of a problem in which we are trying to determine if it is continuous and differentiable at a specified X value. So let's start with the continuity piece. And in order to tackle that one, we're going to utilize our three steps in the test for continuity. So of course the first step is that we need to verify that there is a function value 1x equals 1. So it's the second piece that allows us to substitute in 1. So we do get a y value of negative 9. Second step is that we need to verify the limit exists as we approach 1. Well because it's a piece function, we are going to have to consider both the left and the right limits. So let's do the left side first. So if we're approaching 1 from the left, remember that means through values less than 1. That's the second piece that applies. So in this case, we're just simply going to get the same thing that we had for the function value. If we now consider the limit as we come from the right, it is the first piece that applies for those values greater than 1. That would be 3 times 1 plus 1 and we get 4. So because of this we figure out that the limit does not exist. So it falls apart for continuity and this function is not going to be continuous at x equals 1. Now because of that theorem we studied, that differentiability implies continuity. If we know it is not continuous at x equals 1, therefore we automatically know it is not differentiable at x equals 1. But we're going to go ahead and prove it anyway. Otherwise you can definitely call upon that theorem and really that's the end of it. So you easily could do that and that is perfectly legal to do that. But let's go ahead and just show you how the differentiability piece would work out otherwise. So for this we'll want to actually find our derivative. So the derivative of the first piece was 3. Remember that was when x was greater than 1. Derivative of the second piece was negative 2, that's when x is less than or equal to 1. Well, obviously they're not equal to each other. Really it's because we're looking specifically at 1 and really what the reason is the derivative from the left at 1 does not equal the derivative from the right at 1. The derivative from the left at 1 is the negative 2. The derivative from the right at 1 is 3 and obviously they're not equal. And that's why it ends up not being differentiable at x equals 1.