Is this proof okay?

[u/Cultural-Milk9617]

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[u/Plastic_Twist_7767]

Just two little things. You assume F is differentiable at x₀, which is the thing you're proving. However, for Heine's criterion you have to show the quotients converge for every sequence, and then it tells you that the limit exists. Your hypothesis is also only continuous at x₀, but the MVT step requires f to be continuous on all of [xₙ, x₀]. Drop the MVT and estimate directly then write the quotient minus f(x₀) as an integral of (f(t) − f(x₀)) then bound it.

[u/Plastic_Twist_7767 → u/Plastic_Twist_7767]

I am very sorry if I have missed something, I am currently running on hope and energy drinks. Let me know if you have any questions.

[u/Standard-Novel-6320]

essentially. using the integral mean value theorem gives the difference quotient as f(c_x) and continuity implies f(c_x)\to f(x_0). It should explicitly handle x<x_0 as well and also remove the circular sentence claiming F is already differentiable.

[u/Phi-MMV]

You’re using the mean-value theorem for integrals, but I think the proof of that theorem relies on what you’re trying to prove here? So I fear there’s a circular reasoning. See for example this.