Property 5
Greatest Integer Function
\(x\in\mathbb R\)
Integer that is not bigger than \(\left[\!\left[ x\right]\!\right]=x\)
Example
\(\left[\!\left[3{,}735\right]\!\right]=3\)
\(\left[\!\left[5{,}99\right]\!\right]=5\)
\(\left[\!\left[0{,}63\right]\!\right]=0\)
\(\left[\!\left[-1{,}275\right]\!\right]=-2\)
\(\left[\!\left[-3{,}46\right]\!\right]=-4\)
\(\left[\!\left[4\right]\!\right]=4\)
\(x\in\mathbb R\)
\(\left[\!\left[ x\right]\!\right]=a\qquad a\in\mathbb Z\)
\(\left[\!\left[ x\right]\!\right]=a\qquad a\in\mathbb Z\)
\(\Rightarrow a\le x<a+1\)
\(\left[\!\left[ x+a\right]\!\right]=\left[\!\left[ x\right]\!\right]+a\qquad a\in\mathbb Z\)
\(\left[\!\left[2x\right]\!\right]=\left[\!\left[ x\right]\!\right]+\left[\!\left[ x+\samefrac12\right]\!\right]\)
\(\left[\!\left[3x\right]\!\right]=\left[\!\left[ x\right]\!\right]+\left[\!\left[ x+\samefrac13\right]\!\right]+\left[\!\left[ x+\samefrac23\right]\!\right]\)
\(\left[\!\left[ ax\right]\!\right]\) function is examined at \(\samefrac1a\) intervals.
\(f(x)=\left[\!\left[ g(x)\right]\!\right]\)
If \(g(a)\in\mathbb Z\) "a" is "f" functions critical point.
1.
\(\left[\!\left[ x+2\right]\!\right]=5\)
\(\Rightarrow S.S.=?\)
2.
\(\left[\!\left[\samefrac{4-x}{3}\right]\!\right]=3\)
\(\Rightarrow S.S.=?\)
3.
\(\left[\!\left[\left[\!\left[ x+2\right]\!\right]+2\right]\!\right]=8\)
\(\Rightarrow S.S.=?\)
4.
\(\left[\!\left[ x+1\right]\!\right]+\left[\!\left[ x+4\right]\!\right]=7\)
\(\Rightarrow S.S.=?\)
5.
\(\left[\!\left[ x\right]\!\right]^2-4\left[\!\left[ x\right]\!\right]=0\)
\(\Rightarrow S.S.=?\)
6.
\(0<\left[\!\left[2x-6\right]\!\right]\le10\)
\(\Rightarrow S.S.=?\)