Problem about pointwise convergence.

$\phi_n(x)$ is converging pointwise to $0$ in $K$.That means

For every $\epsilon>0$ and $x\in K$ exists $n_x\in N$ st $\phi_{n_x}(x)<\epsilon$.

And then author says because $\phi_{n_x}$ is continuous then there exists neighborhood of $x$ $\Delta x$ st $t\in K\cap\Delta x$ follows $\phi_{n_x}(t)<\epsilon$.Can you explain how from continuity we got that?If we have that $\phi_{n_x}<\epsilon$ in $K$ why without any continuity does not follow that it is also true for for any subset of $K$ for example $\Delta x\cap K$?

Let $(\phi_n)_{n \in \mathbb{N}}$ be a sequence of continuous functions in $K$. For fixed $x \in K,\varepsilon >0$, we suppose that $\exists N:n\geq N,\,|\phi_n(x)|<\varepsilon$. As the functions are continuous, we have that for $n\in \mathbb{N},\eta >0$ there exists a $\delta >0$ s.t. $t \in V_\delta(x)\implies \phi_n(t)\in V_\eta(\phi_n(x))$. Now let $n=N$, choose $\eta$ s.t. $$V_\eta(\phi_N(x))\in (-\varepsilon,\varepsilon)$$ then there exists a $\delta > 0$ s.t. $t \in V_\delta(x)\implies \phi_N(t)\in V_\eta(\phi_N(x)),\,|\phi_N(t)|< \varepsilon$.