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I'm trying to understand one part of the fundamental theorem of calculus in the Spivak's calculus book. On page 282 he defined:

$$m_h=\inf\{f(x);c\le x\le c+h\}$$

$$M_h=\sup\{f(x);c\le x\le c+h\}$$

Afterwards, he said

$\lim_{h\to0}m_h=\lim_{h\to 0} M_h=f(c)$, since $f$ is continuous.

I understand intuitively why this is true, but I don't know how to prove it rigorously.

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Since $f$ is continuous, there is $x_h \in [c,c+h]$ such that $m_h=f(x_h)$.

With $h \to0$, we get $x_h \to c$. Then, by continuity, we derive $m_h \to f(c)$.

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  • $\begingroup$ Thank you for your answer, but how can we prove rigorously $h\to 0$ implies $x_h\to c$? $\endgroup$ – user42912 Aug 16 '17 at 7:30
  • $\begingroup$ We have $c \le x_h \le c+h$ $\endgroup$ – Fred Aug 16 '17 at 7:44

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