Limits-Continuity+and+Discontinuity

​Continuity and Discontinuity What is “continuity” (in calculus)? The concept is pretty straightforward and almost self explanatory. Basically, a function is considered continuous if its graph doesn’t have any gaps, jumps or asymptotes. If it does have any of these it is discontinuous. A more formal definition is up ahead.

Another way you can determine if a function is continuous is by trying to draw the curve. The curve is continuous if you can draw it without lifting up your pencil like in this example:

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The graph shown above has two loops with no gaps, jumps, or asymptotes present in any of the x-vaues of the function. Thus by the definition mentioned previously in the wiki and by the great ease in being able to draw the function without lifting the pencil off the paper, the function is continuous.

In this next example, the function in the interval of f(-1) and f(1) [the smile] is continuous while the other curves shown are not considered functions because they are the equations of circles.

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As mentioned beforehand, functions are discontinuous if there is one or more of the following in its graph: removable discontinuities, step discontinuities and/or vertical asymptotes. Here are some examples. __Examples:__ ** __**(Removable Discontinuity Example)**__: (x - 3)** This function is not continuous since there is a removable discontinuity at c. [There’s no f(c)]
 * __X^2 - 9__

An easy way to visualize removable discontinuity is to imagine a puzzle near completion except that one piece is missing. This image may help: The last piece is missing! Source: [|www.p-systemsinc.com/]

__**(Removable Discontinuity. Ex. with f(c) elsewhere)**__: This function is not continuous since there is a removable discontinuity at c. [f(c) ≠ the limit of f(x) as x approaches c from both sides]
 * }**

This type of discontinuity can be illustrated with a crate of apples placed on the ground with one apple out of the crate.

Source: http://www.blisstree.com/chocolatebytes/files/2007/10/chocolate_apple_crate.jpg

__**(step discontinuity. Ex.)**__ : This function is not continuous since there is a step discontinuity at c. [There’s no limit of f(x) as x approaches c]

__**(Infinite Discontinuity Example):**__ x**
 * __1__

source: http://www.revisioncentre.co.uk/gcse/maths/1overx.gif This function is not continuous since there is an infinite discontinuity (i.e. vertical asymptote) at c. [There’s no limit of f(x) as x approaches c and there’s no f(c)]

__**(Cusp Continuity Example)**__ : **x^(4/3)** source: http://www.sosmath.com/calculus/diff/der01/der01_2.gif

This function is continuous even though there is a cusp at c. **__CUSPS ARE CONTINOUS__!!!** [f(c) = the limit of f(x) as x approaches c from both sides]

Example: Source: [|www.karlscalculus.org/ pr5_2-3.html]
 * NOTE**-though cusps are continuous in f(x), the derivative taken at cusps are ** __Discontinuous__ **

f(x) has a cusp at f(0) and that can be proven by the derivative f'(x) where at f'(0) there is a step discontinuity.

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**__(Normal Continuity Example):__** **y = x** This function is continuous just because. [f(c) = the limit of f(x) as x approaches c from both sides] As long as you remember these six examples, you should have no problem determining whether a function is continuous or not. Here’s a formal definition to all of you who just have to have one. ======

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 * __When is f(x) continuous on an interval?__ ** f(x) is continuous on an interval for values of x iff it is continuous at every value of x on the interval. At the left endpoint, only the limit of f(c) as x approaches c from the right side (+) must equal f(c). At the right endpoint, only the limit of f(c) as x approaches c from the left side (-) must equal f(c). ** __Cusps are continuous too.__ ** Cusps are points on a graph where the function **is** continuous but **not** differentiable (meaning it has no derivative). ======

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 * __One-sided limits and Piecewise functions.__ ** One sided limits and piecewise functions are an important concept to continuity.They’re fairly simple. One-sided limits are the limits that are approached as x approaches c from the left (-) or right (+). One-sided limits may be different for a point as x approaches c from the left (-) and as x approaches c from the right (+). Piecewise functions sometimes have different limits for f(x) as x approaches c from the left (-) and from the right (+). Therefore, piecewise functions that have different ======

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 * __Property: One-sided limits.__ ** lim f(x) //__means__// x approaches c from the left (through values of x on the “–“ side of c) x => c- lim f(x) //__means__// x approaches c from the right (through values of x on the “+“ side of c) x => c+ L = lim f(x) iff L = lim f(x) = lim f(x) x => c : x => c- : x => c+ This is what a piecewise function looks like and how the corresponding function is written. ======

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__**(Piecewise Graph Example and Piecewise Function Example):**__ ******{{: {{sin(x) :x<-2 and x > 2}}}** } ======