Study of continuous function theory:
From the definition of continuity, we know that f(x) is said to be continuous function at 'c' if it is defined on a interval containing 'c' and The continuity limits define the continuity of a function over those limits. Thus to satisfy the continuity definition, the function should hold true for all this properties.
Function should be defined at 'c'
The limit should exist as x -> c according to the continuity calculus.
The limit of a function and f(c) needs to be equal.
If a graph of a function is not broken anywhere, it satisfies the definition of a continuous function. One another important concept is that of uniform continuity. For a given metric space, the uniform continuity function requires f(x) and f(y) closer to each other provided that x and y are close enough to each other. The equicontinuity of a group of functions is generalization to this concept.
If we have a discrete functions and want to approximate it as continuous functions, a method called continuity correction can be used. Say for example, X is having a Poisson distribution and the value expected is λ, then the variant of X will also be λ and P(X ≤ x) = P(X < x + 1) ≃ P(Y ≤ x + ½). This addition of ½ is called the correction of continuity.
Determine the continuity with continuous function examples:
Lets see some of the continuous function examples and determine the results for them.
(1) Determine the continuity at x = 1 for given function
Here f(x) = 2 for x = 1, i.e. f(1) = 2.
Now limit for x -> 1 is given by L = 3x – 5 = 3*1 – 5 = -2 ≠ f(1). Hence f(x) is not a continuous function.
(2) Determine the continuity at x = 3 for given function
Given that f(3) = 9/2. Now limit for x -> 3 is given by L = (x3 -27)/(x2 – 9). If we simplify it, we get
L = (x-3)(x2 + 3x + 9) / (x-3)(x+3) = (x2 + 3x + 9) / (x+3). Putting 3 in above equation,
L = (32 + 3*3 + 9) / (3 + 3) = 27/6 = 9/2 = f(3). Hence f(x) is a continuous function at x = 3.
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