Monotonicity and boundedness of $F(x)=\max_{t\in [a,x]}f(t)$

Question

Let $f:[a,b)\to \mathbb R$ be a continuous function. Define $F:[a,b)\to \mathbb R$ by $F(x)=\max_{t\in [a,x]}f(t)$. Then state true or false for the following: $1$) $F(x)$ is continuous and necessarily monotonic. $2$) $F(x)$ is necessarily bounded.

Since $f(x)$ is continuous on $[a,x]$ as well, then $F(x)$ is also a continuous function. Here Is the maximum function of a continuous function continuous? is a descriptive solution for the same. When it comes to the $F(x)$ being monotonic and if it is a constant then it is true ( but bounded as well). I think that $F(x)$ will be monotonic (by IVP no any jumps will be there). I am not sure if it is bounded. Need a hint for a better solution. Thanks.

Answer 1

Since you already provide a reference that establishes the continuity of $F$, I'll just gather the comments to provide an answer.

1. The function $f:[0,1)\to\mathbb{R}, \quad x \mapsto \dfrac{1}{1-x}$, for which we have $F(x)=f(x)$, provides a counterexample for the boundness of $f$. Hence, 2) is false.

2. Taking $a \leq x_1 \leq x_2 < b$, we have that $$ F(x_1)=\max_{t \in [a,x_1]}f(t) \leq \max_{t \in [a,x_1]\cup[x_1,x_2]}f(t) = \max_{t \in [a,x_2]}f(t) = F(x_2), $$ which justifies that $F$ is non decreasing.