# Continuity of increasing function

Problem Statement: Show that if $I= [a,b]$ and $f:I \rightarrow \mathbb{R}$ is increasing on $I,$ then $f$ is continuous at $a$ if and only if $f(a) = inf \{f(x): x \in (a,b] \}.$

I'm not exactly sure how to approach this one in particular, however a similar problem is proving that if $f$ is increasing on $I$ and $c$ is not an endpoint, that $\lim_{x \rightarrow c^+} f= inf \{ f(x): x \in I,\ x > c \}.$ And here's how I would prove that one:

$Proof.$ Let $S = \{ f(x) : x \in I, x > c \}.$ Since $c$ is not an endpoint, $S$ is not equal to the zero set. And since $f$ is increasing on $I,\ f(c) \le f(x)$. So $S$ is bounded below by $f(c)$ and therefore the infimum of $S$ must exist, call it $L$. $\forall\ \varepsilon > 0.$ Since $L$ is the infimum of $S$, $L + \varepsilon$ is not a lower bound and there exists $x_\varepsilon \in I,\ x_\varepsilon > c$ such that $$L \le f(c) \le f(x_\varepsilon) < L + \varepsilon.$$ Choose $\delta = x_\varepsilon - c >0$ then $\forall\ x$ satisfying $0 < x - c< \delta\$ i.e. $c< x< x_\varepsilon$
$$L - \varepsilon < L \le f(c) \le f(x) \le f(x_\varepsilon) < L + \varepsilon$$ so then we see that $$|f(x) - L| < \varepsilon$$ implying that $$\lim_{x \rightarrow c^+} f = inf \{ f(x): x \in I, x > c \}.$$

I would appreciate help showing the first problem, preferably in the same style as I did the second one. Thanks in advance.

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