Continuity
Let us now consider the calculus explanation of why a function is continuous. Continuity is a mathematical concept that refers to the ability of a function to be represented as a continuous line on graph. In other words, if you zoom in on any portion of the graph, it looks like a straight line. The most basic example of continuity is when a function has no breaks in it: continuous functions are continuous from left to right and from top to bottom. A function is continuous if it can be drawn without picking up your pencil and without any breaks in the graph. A rational function such as this, which you have already seen, 1s not continuous if the denominator equals 0. f(x)=(3x-1)/(x^2 -4) [Graph] [Graph] x^2 -4=0 x^2=4 x=+-2 Discontinity lim_(x->a) f(x)=3 ———— f(a)=DNE To show continuti: -> lim_(x->a)f(x) exists -> lim_(x->a^-)=lim_(x->a)+f(x) -> lim_(x->a)f(x)=f(a) The graph of function such as this one, which has been created by a piecewise-defined function. is not continuous at the points where the two pieces join. We’ll call this function f, and you can see that it has a hole in the graph at a. So let’s give a calculus-based reason why this function is discontinuous at this point. We have discussed limits thus far In our study of so calculus, so let’s apply that concept here. What is the limit of the function y = f(x) as x approaches a? We can see that the function approaches 3 as x approaches from the left. and also approaches 3 as x approaches from the right. As: result. the limit of this function is 3. If the limit does not exist, and if the limit from the left is different from the limit from the right, then there will be a jump discontinuity and therefore the function will not be continuous. If the limit does not exist, the function is not going to be continuous. It’s not possible, because this is the visual you’ll get every time. In this case, the limit exists, but it is not guaranteed that the y value will be there. Indeed, the function may not be continuous at that point. If I had filled in the hole, then I would have a continuous function. And that gives us one of our two criteria for continuity: that a function must be continuous and its values at a given point must equal its limit. In order for a function to be continuous at a certain point, its limit as we approach that point must equal the same value as its derivative at that point. To show continuity, you need to first show that the limit as x approaches a number of f(x) exists. In order to prove that a limit exists, it is necessary to show that the limit as x approaches a from the left equals the limit as x approaches a from the right. The next step is to show that the limit equals the function value when x=a.
