"So That" vs. "Such That" 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 $x \in \mathrm{Dom}\,(f)$:  $$ |x - c| \le \delta \ \Rightarrow \ |f(x) - f(c)| \le \varepsilon $$

 A: Clauses that begin "such that" function as adjectives; they modify nouns. Clauses that begin "so that" function as adverbs; they modify verbs (or adjectives or other adverbs).
In both of the examples in the question, "such that" is the correct choice, because the clause modifies $\delta$.
In certain cases, either choice can be justified because the difference in meaning ends up not mattering. For example, in "Choose $n$ such that $2n+1$ is prime," the "such that" clause is about $n$. In "Choose $n$ so that $2n+1$ is prime," the "so that" clause is about the action of choosing; it says to do the choosing in a manner that ensures that $2n+1$ is prime. The overall effect of the two options is the same. (Personally, I'd prefer "such that" in this example.)
A: ‘so that’

*

*introduces a clause that (inessentially) conveys the intended consequence of the preceding instruction/assertion

*logical sense: therefore

*means ‘in order that’


*

*Let $f(x)=x^3+3x^2−10x$ so that $f(1)=−6$ and $f(2)=0.$

‘such that’

*

*introduces a (modifying) clause that stipulates a restriction

*logical sense: and

*variously means ‘in which’, ‘satisfying’, ‘for which’


*

*$S$ is the set of reals such that each, for some natural $k,$ equals $2k.$

*Every complex $z$ is such that $|z|$ is nonnegative.

*For each real $x$ such that $f(x)=7,$ we have that $2<x<9.$

*Suppose that $n$ is an integer such that $5|(n + 2).$

*There exists a real $c$ such that $f(c)=0.$

In the OP's given examples, ‘such that’ is correct whereas ‘so that’ is wrong.
