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Let $f: [0, \infty) \rightarrow \mathbb{R}$. Define the value of the left-hand limit of $f$ at $t>0$ to be $f(t^-) = \lim_{x \rightarrow t^-} f(x)$. Define the "left-hand limit function" of $f$ as $f^-: (0, \infty) \rightarrow \mathbb{R}, f^-(t) = f(t^-) = \lim_{x \rightarrow t^-} f(x)$. Prove that $f^-$ is left continuous at each $t>0$.

Let $t>0$. To prove $f^-$ is left continuous at $t$, I need to show that $\lim_{t \rightarrow t^-}f^-(t) = f^-(t) = f(t^-)$.

My question is, how can I use the sequential definition (not the $\epsilon-\delta$ definition) of left continuity to prove this question? In other words, let $(s_n)$ be a sequence contained in $(0, \infty)$ such that $s_n<t$ for all $n$ and $s_n \rightarrow t$. I need to show that $\lim_{n \rightarrow \infty} f^-(s_n) = f^-(t) = f(t^-)$. I have also been given a hint (but I have no idea where to incorporate it):

If $f(t^-)$ exists and is finite, then it is equivalent to: $$f(t^-) = \sup \inf_{s<t} \{f(v): s \le v < t\} = \inf \sup_{s <t} \{f(v) : s \le v < t\} $$

Any help would be greatly appreciated!

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  • $\begingroup$ A working approach (which does not really use the hint): For every $s_n$, there is also a sequence $r_{n,k} \to s_n^-$ for $k\to\infty$. You can use this to pick a sequence $p_n$ such that each $p_n$ is equal to some $r_{n,k}$, we have $p_n \to t^-$ and each $f(p_n)$ is close to $f(s_n)$. $\endgroup$ – mlk Jul 27 '17 at 8:12
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Using the definition of $f^-(t)$, given $\varepsilon>0$ there is $\delta>0$ such that $$|f(s)-f^-(t)|\le\varepsilon $$ for all $t-\delta\le s<t$. Since $s_n\to t^-$, there is $\bar n$ such that $t-\delta<s_n<t$ for all $n\ge \bar n$. Fix $n\ge \bar n$. Then for $t-\delta<s<s_n$ you have $$|f(s)-f^-(t)|\le\varepsilon $$ or equivalently, $$f^-(t)-\varepsilon\le f(s)\le f^-(t)+\varepsilon. $$ Letting $s\to s_n^-$ in the previous inequality, you get $$f^-(t)-\varepsilon\le f^-(s_n)\le f^-(t)+\varepsilon. $$ So you have $|f^-(s_n)- f^-(t)|\le \varepsilon$ for all $n\ge \bar n$.

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