$f(x)=[x]+(x−[x])^{[x]}$ prove that this function is continuous for $x\ge 1/2$ and increase in $[1,+ ∞)$

Question

Consider

$$f(x)=[x]+(x−[x])^{[x]}$$ where $[x]$ is the floor function.

I have to show that $f$ is continuous for $x\geq 1/2$ and increasing in $[1,\infty)$.

The graph of the function is clear but i don't know how to prove that is continuous for $x\geq 1/2$ and increase in $[1,+ \infty)$.

Answer 1

Hint:

Let $x = n + y$ such that $n = [x]$ and $y = \{x\}$. We have $f(x) = n + y^n$.

Prove:

• $f$ is continues when $0<y<1$.
• $f$ is continues when $y=0$. (One needs to consider $0+$ and $0-$.)