%%visits: 2 ## intuition
Proof is in the lecture notes, intuitive proof is here. Idea: Bisection method
Suppose f(a) < f(b). Let d be between f(a) and f(b). Define [a0, b0] ⊃ [a1, b1] ⊃ … so that: 1) $b_n - a_n = \frac{b-a}{2^{n}}$ (It should be halved each time) 2) f(an) < d < f(bn) due to continuity an must converge from the left to a limit bn must converge from the right to a limit As the interval gets smaller, then the limit must equal d.
Corollary of intermediate_value_theorem: If f ∈ C[a, b]
then $\ran f = [m,M]$ where m = inf f and M = sup f in the interval.
## rigour Intermediate value theore := If f ∈ C[a, b]
then f attains every value
between f(a) and
f(b) ## exam clinic
## examples and non-examples ## resources tags
:math:calculus_1:calculus_2:real_analysis: