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I am having some confusions about the removable discontinuities. most of the functions with removable discontinuity I have faced are piece wise functions.example: $f(x)=\begin{cases}x+2,x\neq 1 \\4,x=1\end{cases}$

my first question:how can the discontinuity of a piecewise function be removed??

2nd question:functions with removable discontinuities usually violate the 3rd condition of continuity,which is understandable for piecewise functions.but the function $f(x)=\frac{(x-1)(x-2)}{(x-1)}$ also has a removable discontinuity at $x=1$.but this function violates the very first condition of continuity ($f(a)$ is not defined).should it still be called a removable discontinuity??

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  • $\begingroup$ sorry I'll edit the function.it was not meant to be continuous $\endgroup$ Jun 5, 2016 at 20:14
  • $\begingroup$ and my 1st question was how can the discontinuity of any piecewise function can be removed $\endgroup$ Jun 5, 2016 at 20:15

2 Answers 2

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It doesn't matter which of the three conditions of continuity are violated, as long as the second condition is met while the third is not, we have a removable discontinuity.

If the second is not met, then regardless of the first we have a nonremovable discontinuity.

I'm assuming your conditions are these:

  1. $f(c)$ is defined
  2. $\lim_{x\to c}$ exists
  3. Limit and actual value are identical at $x=c$
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  • $\begingroup$ thanks for your answer.can you answer my first question?? $\endgroup$ Jun 5, 2016 at 20:17
  • $\begingroup$ Because $\lim_{x\to1}f(x)\neq f(1)$ (I.e. because conditions 1 & 2 are met but 3 is not) $\endgroup$ Jun 5, 2016 at 20:32
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A function has a removable discontinuity at $x_0=a$, if and only if the limit for $x\rightarrow a$ exists (left and right limit must coincide!) and if it is not continous at $x_0=a$ (either not defined or with the wrong value).

To remove a discontinuity, either add or change a point (in the first example , change $(1/4)$ to $(1/3)$)

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  • $\begingroup$ thanks for your answer.can you answer my first question?? $\endgroup$ Jun 5, 2016 at 20:17

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