If f:[0,1]-> R is continuous and ∫ f (o to x) = ∫ f (x to 1) for all x ∈ [0,1], show that f(x)=0 for all x ∈ [0,1]
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Note that ∫(t = 0 to x) f(t) dt = ∫(t = x to 1) f(t) dt
Differentiate both sides of ∫(t = 0 to x) f(t) dt = ∫(t = x to 1) f(t) dt via FTC:
This yields f(x) = -f(x) ==> f(x) = 0.
Since x is any element from [0, 1], we are done
Differentiate both sides of ∫(t = 0 to x) f(t) dt = ∫(t = x to 1) f(t) dt via FTC:
This yields f(x) = -f(x) ==> f(x) = 0.
Since x is any element from [0, 1], we are done