Disclaimer: I'm no expert-this is really a question for the analysts and historians of mathematics.

As far as I know,the terminology came into existence in the early 20th century distinguish the classical "calculus" type analysis(hard analysis) from the new point set topology/functional analysis approach (soft analysis). A hard analytic argument uses a direct calculation or construction of an exact estimate bounds of specific function or function types to prove a statement.A soft analytic arguement uses the general topological or geometric properties of a space in which a function or function class is defined to prove a result indirectly without a precisely calculated "bound".

For example, the fact that the Cantor set has measure zero is a "hard" analytic arguement;it uses an epsilon-delta arguement to show the limit of the sequence of "slices" of the lengths of it's component intervals on the real line converges to 0.

An example of a "soft" analytic arguement: (IVT) Let F be a continuous function defined on a connected subset of the real line i.e. an interval with a well defined least upper bound and greatest lower bound. Then the function is defined at every point inbetween the lub and the glb. A soft proof would be as follows: Since an interval I of R is a connected subset of R and f is continuous,then f(I) is also connected. Therefore,for every x in I, f(x) is in f(I). Notice this proof does not involve a direct computation of bounds that proves f(x) is in the image set of f (although it certainly COULD be proven that way).

Anyway,that's how Gerald Itzkowitz taught it to me and I learned a long time ago to trust him on these matters........

Disclaimer: I'm no expert-this is really a question for the analysts and historians of mathematics.

As far as I know,the terminology came into existence in the early 20th century distinguish the classical "calculus" type analysis  (hard analysis) from the new point set topology/functional analysis approach (soft analysis). A hard analytic argument uses a direct calculation or construction of an exact estimate bounds of specific function or function types to prove a statement. A soft analytic arguement uses the general topological or geometric properties of a space in which a function or function class is defined to prove a result indirectly without a precisely calculated "bound".

For example, the fact that the Cantor set has measure zero is a "hard" analytic arguement; it uses an epsilon-delta arguement to show the limit of the sequence of "slices" of the lengths of it's component intervals on the real line converges to 0.

An example of a "soft" analytic arguement: (IVT) Let $f$ be a continuous function defined on a connected subset of the real line i.e. an interval with a well defined least upper bound and greatest lower bound. Then the function is defined at every point inbetween the lub and the glb. A soft proof would be as follows: Since an interval $I$ of $\Bbb R$ is a connected subset of $\Bbb R$ and $f$ is continuous, then $f(I)$ is also connected. Therefore, for every $x \in I$, $f(x)$ is in $f(I)$. Notice this proof does not involve a direct computation of bounds that proves $f(x)$ is in the image set of $f$ (although it certainly COULD be proven that way).

Anyway, that's how Gerald Itzkowitz taught it to me and I learned a long time ago to trust him on these matters........