Help with finding a function, $\lim \ne \inf$

Question

I am looking for an example for a function where:

$$\lim\limits_{x \to a^+} f(x) \neq \inf\, \{f(x) \mid a\leq x \leq b\},\quad a<b,$$

$$f:[a,b]\to\mathbb{R}\;\;\text{ and }\;f\;\text{ is an increasing function.}$$

Can you guys give me a hint of the way of thinking to find such a function? (Notice that $x \to a^{+}$)

The function must be an increasing function.

Thanks!

Answer 1

Actually, you can generate lots of functions satisfying this condition since the function is not supposed to be continuous.

For example, $$f(x) = \begin{cases} 0, \ \ x = 0 \\ \sqrt x + 1, \ \ x \in (0, 1] \ \ \end{cases}$$ Then, we have $$\lim_{x\to0^+} (\sqrt x + 1) = 1 \quad \ne \quad 0 = \inf\{f(x) \mid 0 \leq x \leq 1\}$$

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