# How do i find discontinuity for a function?

##### 1 Answer

Oct 17, 2014

*From a graphical standpoint, a discontinuity on a function will occur at any point where there is either a jump, asymptote or "hole" in the graph. From an analytical standpoint, a discontinuity occurs when any of the following situations is true*:

- For a given point
#a# in the domain of the function (that is, at#x=a# )#lim_(x->a^+)f(x) != lim_(x->a^-)f(x)# (That is, the limit of the function#f(x)# as#x# approaches#a# from the right is not equal to the limit as#x# approaches#a# from the left). This situation is typically called a**jump discontinuity**or**step discontinuity**. An example of a function where this would occur would be a function#g(x)# defined such that#g(x) = 0# for all#x<0# , and#g(x) = 1# for all#x>=0# . Graphically, we will see the function "jump" at the discontinuity. - Alternately, it is possible that either the right-hand or left-hand limit (or possibly both) simply does not exist, or is infinite. These situations are referred to as
**infinite discontinuities**or**essential discontinuities**(or rarely,**asymptotic discontinuities**). On a graph, an infinite discontinuity might be represented by the function going to#+-oo# , or by the function oscillating so rapidly as to make the limit indeterminable. An example would be the function#1/x^2# . As#x->0# from either side, the limit of the function goes to#oo# . For the second type, one may consider#sin(1/(x-1))# , which will begin to oscillate rapidly as we approach#x=1# from either direction, to the extent that we cannot define the limit because even the slightest changes in our#x# value near#x=1# can cause drastic changes in our function. - Finally, there is the situation where both limits exist, are not infinite, and are equal to one another, but not equal to the value of the function
#f(x)# at the given point. These cases are referred to as**removable discontinuities**. Graphically, these will appear like a "hole" in the graph of the function. - In some cases, for the value
#x=a# , the function*will*still be defined, but is simply not equal to the limit (for example, a function defined as#h(x) = x# for all#x!=3# and#h(x)=0# for#x=3# . - In other cases, the function will also be undefined at that point. While some will still classify these cases as removable discontinuities, others will insist on the more precise term of
**removable singularity**, as the function is not simply discontinuous at this point, but nonexistent. An example of this would be the function#j(x) = x^2/x# . By simple division, this will resemble the function#j(x) = x# , but the function will be undefined at#x=0# (since#j(0) = 0/0# , which is undefined.