# Counter-example of continuous Z-periodic functions

In Chapter 16 of Tao's Analysis II, while giving the definition of $$C(\mathbb{R/Z},\mathbb{C})$$, Tao also has add the following one-sentence claim:

By "continuous" we mean continuous at all points on $$\mathbb{R}$$, merely being continuous on an interval such as $$[0,1]$$ will not suffice, as there may be a discontinuity between the left and right limits at $$1$$(or at any other integer).

However, this claim seems to be false, as for a complex-valued $$\mathbb{Z}$$-periodic function $$f$$ that is continuous on $$[0,1]$$, $$f$$ is left continuous on $$1$$ and by periodicity $$f$$ also continues on $$[1,2]$$ thus right-continuous on $$1$$, then the left and right limits are equal, as $$f(1) = f(0)$$. So what's wrong with my justification and how can I find a suitable counter-example?

• Obviously nothing wrong. Perhaps the book meant that $C(\Bbb{R/Z,C})$ can be identified with the functions $[0,1]\to \Bbb{C}$ continuous and such that $f(0)=f(1)$. Commented Feb 6, 2023 at 20:12
• Possibly just a misprint, since on this page, before that sentence, he speaks repeatedly of the half-open interval $[0,1),$ not of $[0,1].$ Commented Feb 6, 2023 at 22:28