Introduction To Discontinuity : Let us first define discontinuity , a function f(x) is said to be continuous in an interval if it is continuous at every point in the domain of f(x). If the interval is closed, say [a, b], the function f(x) is continuous at every point of the interval including the end points a and b. If the interval is open, say ]a, b[, we exclude the end points of the interval. If the function is not continuous at a point, say c, we say that the function is discontinuous at c and c is called a point of discontinuity of f(x) .The graph of the function y = f(x)= 5 for x > 0 and 1 for x = 0, when drawn, we see that the function is discontinuous as there is a jump in the graph at x = 0. If we carefully observe that the function is continuous to the left of O and also to the right of O.
In other words, we say that left hand limit is existing and also the right hand limit is existing but they are not equal. ).Now let us more understand what is discontinuity? The function f is said to be discontinuous at a point a of its domain D if it is not continuous thereat. The point a is then called the point of discontinuity of the function. The discontinuity may arise due to any of the following situations: (i) lim x -> a+ f(x) or lim x -> a- f(x) of both may not exist. (ii) lim x -> a+ f(x) as well as lim x -> a- f(x) may exist, but are unequal. (iii) lim x -> a+ f(x) as well as lim x -> a- f(x) both may exist, but either of the two or both may not be equal to f(a).
We classify the points of discontinuity according to the various situations discussed above:
1. Removable discontinuity: Now let us understand What is a Removable Discontinuity ? A function f is said to have removable discontinuity at x = a if lim x -> a- f(x) = lim x -> a+ f(x) but their common value is not equal to f(a). Such a discontinuity can be removed by assigning a suitable value to the function f at x = a.
2. Discontinuity of the first kind: What is infinite discontinuity ? A function f is said to have a discontinuity of the first kind at x = a if lim x -> a- f(x) and lim x -> a+ f(x) both exist but are not equal. f is said to have a discontinuity of the first kind from the left at x = a if lim x -> a- f(x) exists but not equal to f(a). Discontinuity of the first kind from the right is similarly defined.
3. Discontinuity of the second kind: A function f is said to have a discontinuity of the second kind at x = a if neither lim x -> a- f(x) nor lim x-> a+ f(x) exists. F is said to have discontinuity of the second kind from the left at x = a if lim x -> a- f(x) does not exist. Similarly, if lim x -> a+ f(x) does not exist, then f is said to have discontinuity of the second kind from the right at x = a.
Know more about Free calculus help.
Let I be an open interval let c belongs to I, and let f be a function defined on I except possibly at the point c. The function f has a jump discontinuity at c if the one sided limits of f exists at c but are not equal. The function f is said to have a removable discontinuity at c if lim x -> c f(x) exists, but f(x) is neither not defined or has a value different from the limit. As a specific example, the function g defined by g(x) = x/|x| has a jump discontinuity at 0 since the left hand limit is -1 and the right hand limit is 1. A removable discontinuity can be removed by giving the function the value of the limit at the given point. The function h defined by h(x) = x sin (1/x) has a removable discontinuity at 0; assigning h(0) = 0 makes h a continuous function at 0.
