Limits and continuity theorems
The theorems that make limits and continuous functions behave.
Squeeze theorem
g(x) \le f(x) \le h(x) \implies \lim_{x \to a} f(x) = L
The squeeze (sandwich) theorem: if g ≤ f ≤ h near a and g and h share a limit L, then f has limit L too. Statement, graph and examples.
Intermediate value theorem
f(c) = N \text{ for some } c \in (a, b)
The intermediate value theorem: a continuous function on [a, b] takes every value between f(a) and f(b). Statement, graph, counterexamples and examples.
Extreme value theorem
f(c) \le f(x) \le f(d) \text{ for all } x \in [a, b]
The extreme value theorem: a continuous function on a closed interval [a, b] has an absolute max and min. Statement, graph, counterexamples and examples.
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