In definitions and exercises, I notice that "so that" and "such that" are seemingly used interchangeably. Are they in fact interchangeable, or is one more appropriate for a specific context?
Note: $\mathrm{Dom}\,(f)$ means the domain of $f$.
Example 1:
Suppose that a function $f$ is continuous at a point $c$ and $f(c) > 0$. Prove that there is a $\delta > 0$ $\color{red}{\text{so that}}$ for all $x \in \mathrm{Dom}\,(f)$, $$ |x-c| \le \delta \ \Rightarrow \ f(x) \ge \frac{f(c)}{2} $$
Example 2:
A function $f(x)$ is continuous at a point $c \in \mathrm{Dom}\,(f)$ if and only if for each $\varepsilon > 0$ there is a $\delta > 0$ $\color{red}{\text{such that}}$ for all $x \in \mathrm{Dom}\,(f)$: $$ |x - c| \le \delta \ \Rightarrow \ |f(x) - f(c)| \le \varepsilon $$