Intermediate Value Theorem

Intermediate Value Theorem (IVT)

Let a<bR where f:[a,b]R be continuous. Let yR satisfies either f(a)>y>f(b) or f(a)<y<f(b) then yf([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 yf([a,b]) immediately (use a=f(a)<y<f(b)=b or vice versa, and since yR and in between the f's then yf([a,b])).