In signum and greatest integer function, why does the line starts with a large dot???????

The function f: R → R defined by
 

is called the signum function. The domain of the signum function is R and the range is the set 

{−1, 0, 1}. The graph of the signum function is given below.

In the graph, f(x) is represented along y-axis.

Now, from the definition of signum function, it is clear that value of f(x)=0 only when x=0.

When we take x>0, value of the function becomes 1 and when we take x<0, the value of f(x) 

becomes -1. To represent it on the graph, we use a small circle at x=0,y=1 and x=0, y=-1 to 

make ot clear that the value of function f(x) is 0 at x=0, as we have x>0, value of f(x) becomes 

1 and when x<0, value of f(x) becomes -1.

Greatest Integer Function

The function f: R → R defined by f (x) = [x], x ∈ R assumes the value of the greatest integer, 

less than or equal to x. Such a function is called the greatest integer function. E.g., [x] = 3 

for 3 ≤ x < 4. The graph of the function is given below.

We know that the value of greatest integer function changes at integer points i.e. value of 

greatest integer of 1.0, 1.11, 1.12, 1.16, 1.2, 1.43, ….., 1.99 will be 1 but the value of 2.0 will be 

2 and remain 2 upto 2.99...

That's why we use small circles to show on the graph that the value of function f(x) is not 1 at 

x=2, it is 1 up to those values of x which are just less than2. Similarly, we represent on the 

graph that the value of f(x) is not 2 at x=3, it is 2 for the values of x which are just less than 3.

Hope you will get the point!