removable discontinuity

Question

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??

sorry I'll edit the function.it was not meant to be continuous — sajjad islam, Jun 05 '16 at 20:14
and my 1st question was how can the discontinuity of any piecewise function can be removed — sajjad islam, Jun 05 '16 at 20:15

score 2 · Answer 1 · answered Jun 05 '16 at 19:58

2

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:

$f(c)$ is defined
$\lim_{x\to c}$ exists
Limit and actual value are identical at $x=c$

answered Jun 05 '16 at 19:58

Leonidas Lanier

4,149

thanks for your answer.can you answer my first question?? – sajjad islam Jun 05 '16 at 20:17
Because $\lim_{x\to1}f(x)\neq f(1)$
(I.e. because conditions 1 & 2 are met but 3 is not)
– Leonidas Lanier Jun 05 '16 at 20:32

Peter · Answer 2 · 2016-06-05T20:23:04.367

1

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 $(1/3)$)

edited Jun 05 '16 at 20:23

answered Jun 05 '16 at 20:09

Peter

84,454

thanks for your answer.can you answer my first question?? – sajjad islam Jun 05 '16 at 20:17

removable discontinuity

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