Hey, everyone. Here, we're going to talk about continuity. We've already been dealing with continuity throughout our last several lesson videos. So here, we're really just putting a name to it. A function is continuous at some point c if lim→xc is the same as the function value.
We've seen this happen for quite a few different functions, so we would say that these functions are continuous at those points. Now, we're going to dive a bit deeper into continuity and also talk about what it means for a function to be discontinuous. Let's first take a look at our function on the left here. To find the limit of this function as xc→2, as x gets really, really close to 2 from either side, my function is approaching a y-value of 4.
Now, looking at the actual value of my function when x is equal to 2, it's actually the exact same as my limit; it's also 4. So here, my limit is the same as my function value. This function is continuous at x2. Now, examining our other function, to find the limit as x2→2, it's approaching a y-value of 1.
However, the actual value of the function when x is 2 shows a hole on my graph, thus it's undefined. The limit is definitely not the same as the function value, indicating this function is discontinuous for x2.
By finding our limit and our function value, we're able to determine continuity using its definition. Using a visual tool, if we trace through our graph without stopping at a specified point, then the function is continuous there. For our parabola, as we trace through the function at x2, it's continuous; but another function shows discontinuity as we must stop and continue on the other side of two because of a hole.
Discontinuities occur for rational functions and piecewise functions. For our example problem, we find if functions are continuous at each given value of c by comparing the limit and function value. If these values are the same, the function is continuous; if not, it's discontinuous. For cnegative 2, we find both values are 1, indicating continuity. For c equals 4, the limit does not exist and the function value is 1, showing discontinuity.
Finally, for c equals 1, our function going down to negative infinity signifies an unbounded behavior where the limit does not exist, and the function value is undefined due to an asymptote, confirming discontinuity. With this understanding of continuity, let's continue practicing. Thanks for watching, and I'll see you in the next one.