Continuity and Discontinuity
A function is said to be continuous if it can be drawn without picking up the pencil. Otherwise, a function is said to be discontinuous. Similarly, Calculus in Maths, a function f(x) is continuous at x = c, if there is no break in the graph of the given function at the point. (c, f(c)). In this article, let us discuss the continuity and discontinuity of a function, different types of continuity and discontinuity, conditions, and examples.
Continuity Definition
A function is said to be continuous in a given interval if there is no break in the graph of the function in the entire interval range. Assume that “f” be a real function on a subset of the real numbers and “c” be a point in the domain of f. Then f is continuous at c if
\(\lim_{x\rightarrow c}f(x)= f(c)\)In other words, if the left-hand limit, right-hand limit and the value of the function at x = c exist and are equal to each other, i.e.,
\(\lim_{x\rightarrow c^{-}}f(x)= f(c)=\lim_{x\rightarrow c^{+}}f(x)\),then f is said to be continuous at x = c
Conditions for Continuity
- A function “f” is said to be continuous in an open interval (a, b) if it is continuous at every point in this interval.
- A function “f” is said to be continuous in a closed interval [a, b] if
- f is continuous in (a, b)
- \(\lim_{x\rightarrow a^{+}}f(x)= f(a)\)
- \(\lim_{x\rightarrow b^{-}}f(x)= f(b)\)
Discontinuity Definition
The function “f” will be discontinuous at x = a in any of the following cases:
- f (a) is not defined.
- \(\lim_{x\rightarrow a^{-}}f(x)\) and \(\lim_{x\rightarrow a^{+}}f(x)\) exist but are not equal.
- \(\lim_{x\rightarrow a^{-}}f(x)\) and \(\lim_{x\rightarrow a^{+}}f(x)\) exist and are equal but not equal to f (a).
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Types of Discontinuity
The four different types of discontinuities are:
- Removable Discontinuity
- Jump Discontinuity
- Infinite Discontinuity
Let’s discuss the different types of discontinuity in detail.
Removable Discontinuity
In removable discontinuity, a function which has well- defined two-sided limits at x = a, but either f(a) is not defined or f(a) is not equal to its limits. The removable discontinuity can be given as:
\(\lim_{x\rightarrow a}f(x)\neq f(a)\)This type of discontinuity can be easily eliminated by redefining the function in such a way that
\(f(a) = \lim_{x\rightarrow a}f(x)\)Jump Discontinuity
Jump Discontinuity is a type of discontinuity, in which the left-hand limit and right-hand limit for a function x = a exists, but they are not equal to each other. The jump discontinuity can be represented as:
\(\lim_{x\rightarrow a^{+}}f(x)\neq \lim_{x\rightarrow a^{-}}f(x)\)Infinite Discontinuity
In infinite discontinuity, the function diverges at x =a to give a discontinuous nature. It means that the function f(a) is not defined. Since the value of the function at x = a does not approach any finite value or tends to infinity, the limit of a function x → a are also not defined.
Continuity and Discontinuity Examples
Go through the continuity and discontinuity examples given below.
Example 1:
Discuss the continuity of the function f(x) = sin x . cos x.
Solution:
We know that sin x and cos x are the continuous function, the product of sin x and cos x should also be a continuous function.
Hence, f(x) = sin x . cos x is a continuous function.
Example 2:
Prove that the function f is defined by \(f(x)= \left\{\begin{matrix} x sin\frac{1}{x} & x\neq 0\\ 0 & x=0 \end{matrix}\right.\) is continuous at x =0
Solution:
Left hand limit at x = 0 is given by
\(\lim_{x\rightarrow 0^{-}}f(x) = \lim_{x\rightarrow 0^{-}}xsin\frac{1}{x}=0\) [Since -1Thus, \(\lim_{x\rightarrow 0^{-}}f(x) = \lim_{x\rightarrow 0^{+}}f(x) = f(0)\).
Hence, the function f(x) is continuous at x =0.
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