[Music] we will start this topic by eyeing the following two graphs of some functions you can pause the video right now quickly run your eyes over them and then hear me out describing each one of them individually intuitively it's easy to tell that this is a continuous function and the other one isn't because its graph is disconnected and yes that's basically how we can picturize a continuous and a discontinuous function a function continuous everywhere in the domain will be a smooth connected curve a nice kind of a mnemonic to think of the graph of a continuous function is that it can be drawn without having to lift a pencil on the paper as long as the pencil is in contact with the paper we can trace a continuous curve what about the graph of a discontinuous function in the graph of the discontinuous function we will find gaps or perhaps a better word would be breaks in the graph the graph is all tidy and connected from the left to the right until we reach here where it breaks and restarts from some place below it then it rises higher above to this point before breaking again and so on let's study this graph in short these are the points on the x-axis that is the domain of the function where the function is discontinuous we call them points of discontinuity one more interesting thing to observe is this portion of the graph consists of just one point so at x3 the graph actually breaks twice you will see this in many discontinuous functions so don't get too surprised moreover we also find these little filled in circles in the corners and also there are unfilled circles above or below filled ones what do they represent before revealing what they represent let me divert your attention to some other fact i will state an important point about continuity to clear a common misunderstanding a function is continuous or discontinuous only in the domain of the function that is only at those x's where the function is defined have a look at this function the graph is as shown there appears to be a big gap in the graph between the numbers 2 and 4 on the x-axis in fact there is no plot between these two points as the function is not defined for the numbers between 2 and 4. the function is only defined for x's between negative 1 and 2 including at negative 1 and 2 and also between 4 and 6 including both 4 and 6. so what can we say about the points between 2 and 4 are they also the points of discontinuity a general perception that builds in our mind when we hear the word discontinuous makes us see any sort of gap in the graph as discontinuity but that's incorrect although this function is continuous in each subdomain taken separately we cannot define continuity anywhere between these two points as they are not in the domain of the function hence earlier that was the reason why i was refraining from using the word gap as it might insinuate a horizontal gap there is rarely a function you will see in the topic with such huge gaps a better way to imagine points of discontinuity is the points where the graph is jumping either up or down and these circles are made to mark the position where the function is defined at those points of discontinuity and where it isn't for instance at point x1 you can see without any sort of mark it will be difficult to tell if the point with coordinates x1 comma f of x1 is connected with the left portion of the graph or with the right in our language we call them holes holes are made on the curve at a point to indicate a break in the graph and a hole is filled into the mark where the function is defined at that particular x so this is where f of x one is defined this is one type of discontinuous function and it shouldn't be surprising now that this type is called jump discontinuity there are two other types of discontinuities removable discontinuity and infinite discontinuity but we will leave them for some other video before ending this video i will leave you with a question to ponder upon look at this graph it kinda looks similar to the graph of the type removable discontinuity which is saw now so the question is is this function continuous at x equal to negative one share your answers with your reasons in the comment section below also do subscribe to our channel to stay connected and as always stay smart stay curious [Music]