Equation of line with a constant increase
We have points (-4,3) and (4,8)
Slope = [8-3] / [ 4 - -4] = 5/8
Equation is
y = (5/8)(x - 4) + 8
y = (5/8)x - 20/8 + 64/8
y = (5/8)x + 44/8
y = (5/8)x + 11/2
Equation of line with constant decrease
We have points (4,4) and ( 9,1)
Slope = [ 1 - 4] / [ 9 - 4] = -3/5
Equation is
y = (-3/5)(x - 4) + 4
y = (-3/5) + 12/5 + 4
y = (-3/5)x + 32/5
Based on your graph, we would have this :
1 if -inf < x -4
f(x) = (5/8)x + 11/2 if -4 ≤ x < 4
(-3/5)x + 32/5 if 4 < x ≤ 9
1) and 2)
This is still "piecewise"......if we can draw a graph without lifting our pencil, it is continous (although it could still be piecewise-continous).....your graph is "broken"...so...it is not piecewise continuous...
3)
This graph is continuous from (- infinity, 4)
It is also continuous from (4, 9 ]
Note that at x = 4....the continuity is "broken"
Also note that the closed circle at x = - 4 "picks up" the continuity from negative infinity
Hectictar can probably develop this even more !!!!