Question

Define a piecewise function on the intervals (-∞,2), (2,5), and [5,∞) that does not "jump" at 2 or 5

Answer 1

Seeing no restriction that says otherwise, a cheap way to do it is to make all three pieces the same function.
That's kinda boring, though.

Answer 2

Also, that first interval would have to be (-∞,2] or else there will be a jump at x=2.

Answer 3

The easiest way is to pick a function that is amiable for the middle segment, and let the lower segment be the value of the middle function at x=2, and the upper segment the value of the middle segment at x=5.

Answer 4

y=2
y=x
y=5

Answer 5

I'd probably put whatever functions I wanted for the outer domains - maybe a quadratic in the left zone, a cubic in the right zone, say, then find what point the quadratic gets to at x=2 and what point the cubic gets to at x=5, then use those two points and put a line in between them.
Ya' know, just to be creative.

Answer 6

each one on the corresponding interval

Answer 7

Example: choose an amiable function for the middle section. I choose f(x)=x^2. f(2)=4 and f(5)=25. So my piecewise function is g(x)=4 for x in (-infty,2); x^2 for x in [2,5];25 for x in (5, infty).

Answer 8

There are dozens of ways to do this, probably some easier than my method. The key is to get the function values at the endpoints to match.

Answer 9

This is the graph of the function that I posted above

Answer 10

\[
f(x)=\begin{array}{cc}
\{ &
\begin{array}{cc}
2 & x\leq 2 \\
x & 2<x<5 \\
5 & x>5 \\
\end{array}
\\
\end{array}


\]

Answer 11

the last x >=5

Answer 12

eliassaab and I used essentially the same method.

Answer 13

Yes. There are an infinite number of ways of doing that.
In fact y=x for any x will do it.

Answer 14

May be for piecewise, we have to change the function on each interval.

Answer 15

thanks for the help !

Answer 16

yw