Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set. In particular, $M$ can be described by a finite set of polynomial equalities and inequalities. Let $\delta_0 > 0$ be a real number.

Suppose that the following condition holds: for every finite collections $t_1,t_2,\cdots,t_K \in [0,1]$ there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x_{t_k}\| \geq \delta_0$ for every $k\in \{1,2,\ldots,K\}$.

I would like to deduce that there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x_{t}\| \geq \delta_0$ for every $t \in [0,1]$.

Is this true?

Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set. In particular, $M$ can be described by a finite set of polynomial equalities and inequalities. Let $\delta_0 > 0$ be a real number.

Suppose that the following condition holds: for every finite collections $t_1,t_2,\cdots,t_K \in [0,1]$ there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x(t_k)\| \geq \delta_0$ for every $k\in \{1,2,\ldots,K\}$.

I would like to deduce that there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x(t)\| \geq \delta_0$ for every $t \in [0,1]$.

Is this true?

Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set. In particular, $M$ can be described by a finite set of polynomial equalities and inequalities. Let $\delta_0 > 0$ be a real number.

Suppose that the following condition holds: for every finite collections $t_1,t_2,\cdots,t_K \in [0,1]$ there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x_{t_k}\| \geq \delta_0$ for every $k\in \{1,2,\ldots,K\}$.

I would like to deduce that there is a continuous function $x : [0,1] \to \mathbb{R}^n_+$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x_{t}\| \geq \delta_0$ for every $t \in [0,1]$.

Is this true?

Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set. In particular, $M$ can be described by a finite set of polynomial equalities and inequalities. Let $\delta_0 > 0$ be a real number.

Suppose that the following condition holds: for every finite collections $t_1,t_2,\cdots,t_K \in [0,1]$ there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x_{t_k}\| \geq \delta_0$ for every $k\in \{1,2,\ldots,K\}$.

I would like to deduce that there is a continuous function $x : [0,1] \to \mathbb{R}^n$ that satisfies:

1. The graph of $x$ is a subset of $M$.

2. $\|x_{t}\| \geq \delta_0$ for every $t \in [0,1]$.

Is this true?