let ab subset R be a closed bound interval let c element ab

let [a,b] subset R be a closed bound interval let c element (a,b) and let f:[a,b] -> R be a function. Suppose that f|[a,c] and f|[c,b] both satisfy the conclustion of the intermediate value theorem. Prove that f satisfies the conclusion of the intermediate value theorem

Solution

f is defined on the the closed interval [a,b]

Since f satisfies the intermediate value theorem in [a,c] f is continuous in [a,c]

so left limit of f(x) as x tends to c = f(c)

Similarly since f satisfies intermediate value theorem in [c,b] f is continuous in [c,b]

so left limit of f(x) as x tends to c = f(c)

Then there is no other possibility than f(c) = left limit = right limit

So f is continuous at c also both the sides

Hence in the interval [a,c)U(c,c)U(c,b] = [a,b], f satisfies intermediate value theorem