William W. answered • 03/07/20
Experienced Tutor and Retired Engineer
f(x) = [[x]] is typically the greatest integer function or sometimes referred to as the floor function. It is a step function in that the value is the integer less than or equal to the value of x.
Example: [[3.24]] = 3 or [[-6.7]] = -7. The graph is a set of steps that are 1 unit wide.
In your case, g(x) = [[x]] + [[5 + x]] let's consider what happens at x = 0 as x increases.
g(0) = [[0]] + [[5 + 0]] = 0 + 5 = 5
g(0.1) = [[0.1]] + [[5 + 0.1]] = 0 + 5 = 5
g(0.5) = [[0.5]] + [[5 + 0.5]] = 0 + 5 = 5
g(0.999) = [[0.999]] + [[5 + 0.999]] = 0 + 5 = 5
g(1) = [[1]] + [[5 + 1]] = 1 + 6 = 7
g(1.999) = [[1.999]] + [[5 + 1.999]] = 1 + 6 = 7
g(2) = [[2]] + [[5 + 2]] = 2 + 7 = 9
We would graph it like this:
So we can see that this function is not continuous. However, it does have pieces that are continuous and so if you were going to take the limit as x approaches "a" and "a" happened to be 2.5 for instance, then you COULD restrict the domain in such a way that the limit would exist. However, if a = 4, the value of the limit does not converge toward a single value. If you are coming in from the left, the value of g(x) as x approaches 4 would be 11. If you were coming in from the right, the value of g(x) as x approaches 4 would be 13. I hope that helps.
