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- Removable discontinuities can be "filled in" if you make the function a piecewise function and define a part of the function at the point where the hole is. In the example above, to make f(x) f (x) continuous you could redefine it as: f(x) = {(x+2)(x+1) x+1, 1, x ≠ −1 x = −1 f (x) = {(x + 2) (x + 1) x + 1, x ≠ − 1 1, x = − 1
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How do you remove a discontinuity from a function?
What is a removable discontinuity?
How do you determine a removable discontinuity?
What if a function f(x) has a nonremovable discontinuity?
In contrary to the removable discontinuity, a function f(x) has non removable discontinuity at x = a if the limit limₓ → ₐ f(x) does not exist. There are two types of nonremovable discontinuities: Jump Discontinuity; Infinite Discontinuity; Jump Discontinuity
Mathematically, for a function \(f(x)\) with a removable discontinuity at \(x = a\), we can say: - \(f(a)\) is undefined or does not equal the limit as \(x\) approaches \(a\). - However, \(\lim_{x \to a} f(x)\) exists. A classic example of removable discontinuity can be represented by the function: \[ f(x) = \frac{x^2 - 1}{x - 1} \]
#f(x)# has a removable discontinuity at #x=a# when #lim_{x to a}f(x)# EXISTS; however, #lim_{x to a}f(a) ne f(a)#. A removable discontinuity looks like a single point hole in the graph, so it is "removable" by redefining #f(a)# equal to the limit value to fill in the hole.
Aug 22, 2015 · Sure, you could have $$ f(x) = \begin{cases} 0 & \text{when }x<0 \\ 1 & \text{when }x=0 \\ 0 & \text{when }0<x<1 \\ 1 & \text{when }1\le x \end{cases} $$ which has a removable discontinuity at $0$ and a nonremovable one at $1$.
If $c$ is a hole or a point discontinuity and the limit of $f$ at $c$ exists, then $c$ is a removable discontinuity. If $c$ is a hole or a point discontinuity and both the left and right limits of $f$ as $x$ approaches $c$ exist but are not equal, then $c$ is a jump discontinuity.
Nov 24, 2023 · This step effectively ‘fills the hole’ or ‘bridges the gap’ in the function at the discontinuity. Redefining Functions with Removable Discontinuities. For Undefined Points: If \( f(a) \) was originally undefined, redefine ( f(a) ) to be the limit value. For example, redefine \( f(x) = \frac{(x^2 – 1)}{(x – 1)} \) at \( x = 1 \) to ...
3 days ago · A real-valued univariate function f=f (x) is said to have a removable discontinuity at a point x_0 in its domain provided that both f (x_0) and lim_ (x->x_0)f (x)=L<infty (1) exist while f (x_0)!=L.
