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Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) ?$$ I am trying to prove this by using the Sobolev embedding theorem:

If $s > n/2$ then there exists $c>0$:$$ \| g \|_{L^\infty(\Bbb R^n)} \leqslant c \| g \|_{W^{s,2}(\Bbb R^n )} \; \text{and} \; g \text{ is continuous}$$ for $g \in W^{s,2} (\Bbb R^n )$.

Here $f \in C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ means that $\| f(t) \|_{W^{s-1,2} (\Bbb R^n )}$ is continuously differentiable on $[0,\infty)$, $C_b^k$ means that $k$ times continuously differentiable functions with bounded derivatives up to order $k$, and $W^{s,p}$ means the usual Sobolev space.

You may use the additional condition $f \in C^\infty ([0,\infty) \times \Bbb R^n )$ if you need.

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