Question

How do i find discontinuity for a function?

Answer 1

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.