# Generality versus core insight trade-off in theorem statements

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For example, consider the calculus:Intermediate value theorem . It's possible to state the theorem as saying the function attains each value between $f(a)$ and $f(b)$. But this introduces an extra parameter which takes up more working memory. It is simpler to state it as saying the function equals zero somewhere. Similarly, we could let $a$ and $b$ be any real numbers, but we could restrict the theorem to the case $a < b$, which is where all of the interesting stuff happens.
for any $t$ between $f(a)$ and $f(b)$, there is $c$ in $[a,b]$ such that $f(c) = t$ there is $c$ in $[a,b]$ such that $f(c) = 0$ $a,b$ are arbitrary real numbers (including the case $a > b$, which makes $[a,b]$ empty so the theorem becomes vacuous, and also including the case $a=b$ so the function consists of a single point, which makes the theorem trivial) $a,b$ are real numbers such that $a $f(a)$ and $f(b)$ can be anything (we don't add hypotheses concerning their values) $f(a) < 0$ and $f(b) > 0$