For all define

where denotes the greatest integer less than or equal to . Draw the function for each of the following on the interval :

- .
- .
- .
- Graph of .
- Graph of .
- The graph of .
- The graph of .

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#
Stumbling Robot

A Fraction of a Dot
#
Draw the graphs of some step functions

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6 comments

### Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):

For all define

where denotes the greatest integer less than or equal to . Draw the function for each of the following on the interval :

- .
- .
- .
- Graph of .
- Graph of .
- The graph of .
- The graph of .

I think you shall define the value of function when x = 2.

a) is wrong. For example h(-0.5) = [-0.5] + [-1] = 0 – 1 = -1.

In reality, the drawing will be like this:

h( -1 ) = -3;

h( (-1, 0.5] ) = -1;

h( (-0.5, 0] ) = 0;

h( (0, 0.5] ) = 2;

h( (0.5, 1] ) = 3;

h( (1, 1.5] ) = 5;

h( (1.5, 2] ) = 6.

(I hope such a record does not create difficulties)

I’m sorry, it’s not. My function is the smallest integer >= x

c) is wrong. Note that ceil(0) * ceil(anything) = 0. So the interval [-1,0] on graph c should be 0.

Let’s take x = -1/2 for instance. We have [x] = -1 and [2x] = -1. Therefore, h(x) = f(x)g(x) = 1

There is a mistake: the interval is closed on the right: you should also include the point at x=2 in the plots