Single point function is not a continuous function according to the intermediate value theorem

Question

According to the intermediate value theorem, we can infer that a single point function, for example $f(x) = \sqrt x + \sqrt(-x)$, which has a range of only $0$ and a domain of only $0$ is not a continuous function. Is my conclusion right or there is actually exception when it is continuous? or outside of the calculus realm of thinking single point function CAN BE continuous?

Answer 1

Although there are similar duplicates here a quick reasoning:

• You consider a real-valued function with domain $D = \{0\}$, so you can use the limit definition of continuity: $$f \mbox{ is continuous at } x_0 = 0 $$ $$\Leftrightarrow \lim_{n\to\infty}f(x_n) = f(0) \mbox{ for all sequences } \{x_n\}_{n \in D}\subset D \mbox{ with } \lim_{n\to\infty}x_n = 0$$
• The only possible sequence converging to $0$ in the domain $\{0\}$ is the constant sequence $x_n = 0$ for all $n \in \mathbb{N}$. $$\lim_{n\to\infty}f(x_n) = f(0) = 0$$ So, $f$ is contiuous.