Let a<b∈R where f:[a,b]→R be continuous. Let y∈R satisfies either f(a)>y>f(b) or f(a)<y<f(b) then y∈f([a,b]).
Proof
Using Continuous Image of Connected Set is Connected, since [a,b] is a connected set, then f([a,b]) is connected. Using Condition for Connected (which we get immediately from the givens), then y∈f([a,b]) immediately (use a=f(a)<y<f(b)=b or vice versa, and since y∈R and in between the f's then y∈f([a,b])).
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