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?