= Solution
Apply part (i) to the continuous difference $h=f-g$. Every <Fourier coefficient> of $h$ vanishes, so every <Fourier partial sum> is zero. The <orthogonal projection> convergence from part (i) therefore gives $\|h\|_2=0$.
If $h(t_0)\ne0$, <continuity> gives an interval on which $|h|$ is bounded below by a positive number. That interval would contribute positively to $\|h\|_2^2$, a contradiction. Hence
$$
\boxed{f=g\quad\text{at every point of the circle}.}
$$
The continuity hypothesis upgrades equality almost everywhere to pointwise equality.
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