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Given a piecewise function with a breakpoint, how can you show that the function is continuous.

For a function to be continuous at a point $x=a$, the following should hold: $\lim_{x\to a} f(x) = f(a)$

However, at a breakpoint $a$ when you approach the limit of $a$ from left ($a^-$) and right ($a^+$), it is not the same (slope is different). This is a necessary condition for a function to be continuous at a point $x=a$. How is this possible?

Example:

$f(x) = \begin{cases} -x-3 &\text{if $x \leq -3$;} \\ x+3 &\text{if $x >-3$}. \end{cases}$

Math98
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  • It's not continuous, for the reason you mention. Nothing to do with slope, of course. The values of the function on one side aren't near the values on the other side. the slopes (if they exist at all) might match on both sides, or they might not. – lulu Jul 02 '23 at 19:59
  • Perhaps you are confused by the phrase "piecewise continuous"? Such a function is generally not continuous. The name refers to functions that are continuous, away from discrete discontinuities. See this question for further discussion. – lulu Jul 02 '23 at 20:02
  • @lulu there is no discontinuity, Wikipedia also mention that it is continuous. I added an example. – Math98 Jul 02 '23 at 20:07
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    That is continuous. $f(3)=0$ and that is the limit on either side. Again, the fact that the slopes do not match is irrelevant. The function is not differentiable at $x=-3$ but that's not what was asked. – lulu Jul 02 '23 at 20:10
  • So, if you approach the limits from both sides ($3^-$ and $3^+$), you will get the same answer, despite the slopes don't match? I thought if the slopes don't match the limits if you approach them from left and right then also don't match. I think that was my confusion. Can you confirm it? – Math98 Jul 02 '23 at 20:16
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    The slopes have nothing to do with it. If $\epsilon$ is a small positive number then $f(-3+\epsilon)=\epsilon$ and $f(-3-\epsilon)=\epsilon$ so we are approaching $0$ on both sides. Again, the function is not differentiable at $x=-3$, precisely because the slopes do not match. But that is a different question. – lulu Jul 02 '23 at 20:25

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The function that you showed is not continuous because it looks like two separate lines which don't ever connect. There are three main types of discontinuity: point, jump, and infinite. Point discontinuity, as said in the name, is when a function is not defined for a point. Jump discontinuity is the type of discontinuity your piecewise function has. The function jumps from one point to another. Infinite discontinuity is when a function has one or more asymptotes. Here are all three examples of discontinuity. enter image description here

Berny
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