What is continuity and how do we determine if a function is continuous or discontinuous at a point on its graph?

A function is said to be [i]continuous at a point[/i] [math]x=c[/math] if [i]all[/i] of the following are true:[br][list][br][*]The function's [math]y[/math]-value approaches some number [math]L[/math] as [math]x[/math] approaches [math]c[/math] from the [i]left[/i];[br][*]The function's [math]y[/math]-value approaches this same number [math]L[/math] as [math]x[/math] approaches [math]c[/math] from the [i]right[/i];[br][*][math]f(c)[/math], the function's value [i]at[/i] [math]x=c[/math], is equal to [math]L[/math].[br][/list][br]If one or more of these requirements is not met, we say the function is [i]discontinuous[/i] at [math]x=c[/math], or that there is a [i]discontinuity[/i] at [math]x=c[/math].[br][br]What do we mean by [math]x[/math] approaches [math]c[/math] from the right or left? Suppose that we want to examine the graph at [math]x=-3[/math]for continuity. Drag the red dot on the [math]x[/math]-axis so that it is placed at [math]x=-3[/math]. We want to observe what happens along the graph as [math]x[/math] "sneaks up" on [math]-3[/math] from the left. Check only the "Show L" box on the right side of the app. A small blue dot to the left of the red appears on the [math]x[/math]-axis. This is our "sneaking" [math]x[/math] value. The left slider at the right of the app will allow you to slide the blue dot closer and closer to the red dot ([math]x=-3[/math]). Follow the blue dotted line up to the graph to see the point for that [math]x[/math]-value, then follow the arrow to the [math]y[/math]-axis to see the [math]y[/math]-value. When the blue dot is as close to the red dot as you can make it, we say we have reached [i]the limit as [math]x[/math] approaches [math]-3[/math] from the left, of [math]f(x)[/math][/i], whose value is the [math]y[/math]-value there, [math]y=3[/math]. We would write this as [math]\lim_{x \to -3-}f(x)=3[/math]. Notice the "[math]-[/math]" following [math]-3[/math]; it indicates "from the negative side (the left)".[br][br]Now check the "Show R" box and repeat, approaching [math]x=-3[/math] from the right this time. Again, it appears that [math]y[/math] approaches [math]3[/math]. In this case, [math]y[/math] went to the same value regardless of which way we approached [math]x=-3[/math]. But once we get to exactly [math]x=-3[/math], there is no point waiting for us; only an empty hole. We would write this approach as [math]\lim_{x \to -3+}f(x)=3[/math]. Notice the "[math]+[/math]" following [math]-3[/math]; it indicates "from the positive side (the right)".[br][br]We will look at three types of discontinuity:[br][br]A [i]removable[/i] discontinuity exists when the function approaches the same [math]y[/math]-value ([math]L[/math]) from both sides of [math]x=c[/math], BUT there is [i]NO[/i] point connecting the two there. There is a "hole" in the graph - as if someone plucked out a single point from an otherwise continuous graph. It is called a removable discontinuity because "filling the hole" fixes the discontinuity. This is done by adding the definition that [math]f(c)=L[/math]. In the graph above, a removable discontinuity exists at [math]x=-3[/math], which could be removed (filled in) by letting [math]f(-3)=3[/math].[br][br]A [i]jump[/i] discontinuity exists when the function approaches [i]different and finite[/i] [math]y[/math]-values from the two sides of [math]x=c[/math]. In the graph above, there is a jump discontinuity at [math]x=-1[/math], where [math]y[/math] approaches [math]1[/math] from the left, but [math]y[/math] approaches [math]-3[/math] from the right, as [math]x[/math] approaches [math]-1[/math].[br][br]An [i]infinite[/i] discontinuity exists when the function approaches [math]+ \infty[/math] or [math]- \infty[/math] on either or both sides of [math]x=c[/math]. In the graph above, there is an infinite discontinuity at [math]x=2[/math], because [math]y[/math] approaches [math]+ \infty[/math] from both the right and from the left. As long as [math]y[/math] is approaching [math]\pm \infty[/math] on at least one side, it is an infinite discontinuity.[br][br]If all three requirements are met, we can say that the function approaches the same [math]y[/math]-value from both sides of [math]x=c[/math], AND there is a point connecting the two. The function is [i]continuous at [/i][math]x=c[/math]. This is true for all other points on the graph not mentioned above.