# Pointwise convergence of a sequence of continuous monotone functions to a continuous function is uniform?

Let $f_n$ be a sequence of continuously differentiable real-valued functions converging pointwise to a continuous function $f$. Assume further that each of the $f_n$, and $f$ itself, is a strictly monotone increasing function (note: the sequence is not necessarily monotone in $n$).

Does this imply that $f_n$ uniformly converges to $f$? And that $f_n$ is therefore equicontinuous?

• You may want to check the second one of Dini's theorems. (Not the first, which is about a monotone sequence of functions; the second [Warning: link towards a French page] is on a sequence of monotone functions). – Clement C. Nov 3 '16 at 14:39
• Do you have a reference? – Nadia M Nov 3 '16 at 14:43
• Quoting this page: "In their book "Regular Variation", Bingham, Goldie, and Teugels mention Polya's extension of this result (Polya's asserting locally uniform convergence when the common domain of the functions is (0,infinity)); see exercise 22 on page 60 of that book. They refer to vol. II of "Problems and Theorems in Analysis" by Polya and Szego, specifically problem 127 of the 1976 edition of the book." (I assume learning French to parse the Wikipedia reference is out of the question). – Clement C. Nov 3 '16 at 14:45
• I know some French, actually. I'll look for that. – Nadia M Nov 3 '16 at 14:46
• Just look for the reference at the bottom of the page. – Clement C. Nov 3 '16 at 14:47

This is false on $[0,1).$ Consider $f_n(x) = x + x^n,f(x) = x.$