Prob. 4 (a), Sec. 20 in Munkres' TOPOLOGY, 2nd edition: Which of these functions are continuous in these topologies?

Question

Let $\mathbb{R}^\omega$ denote the set of all infinite sequences of real numbers. Let the uniform metric $\tilde{\rho}$ on $\mathbb{R}^\omega$ be defined as follows: $$\tilde{\rho}(x,y) \colon= \sup \left\{ \ \min \left( \ \vert x_i - y_i \vert \ , 1 \ \right) \ \colon \ i = 1, 2, 3, \ldots \ \right\} \ \ \ \forall x = (x_i)_{i \in \mathbb{N}}, \ y=(y_i)_{i \in \mathbb{N}} \in \mathbb{R}^\omega.$$ Then the topology induced by this metric is called the uniform topology on $\mathbb{R}^\omega$.

And, the box topology on $\mathbb{R}^\omega$ is the one havaing as a basis all sets of the form $$ (a_1, b_1) \times (a_2, b_2) \times (a_3, b_3) \times \cdots, $$ where $\left(a_i \right)_{i \in \mathbb{N}} \in \mathbb{R}^\omega$ and $\left(b_i \right)_{i \in \mathbb{N}} \in \mathbb{R}^\omega$ are such that $a_i < b_i$ for each $i = 1, 2, 3, \ldots$, and $(a_i, b_i)$ denotes the segment (i.e. open interval) with $a_i$ as the left endpoint and $b_i$ as the right endpoint.

Now with respect to which of the above two topologies on $\mathbb{R}^\omega$ are the following functions from $\mathbb{R}$ to $\mathbb{R}^\omega$ continuous? $$f(t) \colon= (t, 2t, 3t, \ldots),$$ $$g(t) \colon= (t, t, t, \ldots), $$ $$h(t) \colon= (t, \frac{t}{2}, \frac{t}{3}, \ldots).$$

My effort:

The functions $f$, $g$, and $h$ are not continuous when $\mathbb{R}^\omega$ is given the box topology. The inverse image under each of $f$, $g$, and $h$ of the basis element $$B \colon= \left( -1, 1 \right) \times \left(-\frac{1}{2^2}, \frac{1}{2^2} \right) \times \left( -\frac{1}{3^2}, \frac{1}{3^2} \right) \times \cdots, $$ for example, contains the point $t = 0$ since $$f(0) = g(0) = h(0) = (0, 0, 0, \ldots) \in B.$$ So, in order for this inverse image to be open in $\mathbb{R}$ with the usual topology, there must be an open interval $\left(-\delta_f, \delta_f \right)$, $\left( -\delta_g, \delta_g \right)$, and $\left( -\delta_h, \delta_h \right)$, for some positive real numbers $\delta_f$, $\delta_g$, and $\delta_h$, respectively, such that $$ \left(-\delta_f, \delta_f \right) \subset f^{-1}(B),$$ $$\left( -\delta_g, \delta_g \right) \subset g^{-1}(B), $$ $$\left( -\delta_h, \delta_h \right) \subset h^{-1}(B). $$ In particular, we must have $$f\left( \frac{\delta_f}{2} \right) \in B,$$ $$g\left( \frac{\delta_g}{2} \right) \in B,$$ $$h\left( \frac{\delta_h}{2} \right) \in B.$$ So, for each $n \in \mathbb{N}$, we have $$\frac{n \delta_f}{2} \in \left( - \frac{1}{n^2}, \frac{1}{n^2}\right), \ \frac{\delta_g}{2} \in \left( - \frac{1}{n^2}, \frac{1}{n^2}\right), \ \frac{\delta_h}{2n} \in \left( - \frac{1}{n^2}, \frac{1}{n^2}\right), $$ and hence $$n^3 < \frac{2}{ \delta_f } \ \mbox{ for all} \ n \in \mathbb{N},$$ $$n^2 < \frac{2}{ \delta_g } \ \mbox{ for all} \ n \in \mathbb{N},$$ $$n < \frac{2}{ \delta_h } \ \mbox{ for all} \ n \in \mathbb{N},$$ each of which is impossible.

Am I right?

That $g$ is not continuous was also shown by Munkres himself in Example 2, Sec. 19 on page 117.

What is the situation for the uniform topology? That is, which of these three functions is continuous relative to the uniform topology?