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The following is from Duoandikoetxea's Fourier Analysis.

Let $X=C(\mathbb{T})$, i.e., continuous period functions (period of 1) with the norm $\Vert \cdot \Vert_\infty$ and let $Y=\mathbb{C}$. Define $T_N:X \to Y$ by $$T_N f=S_N f(0)=\int_{-1/2}^{1/2}f(t)D_N(t)dt.$$ Define the Lebesgue numbers $L_N$ by

$$L_N =\int_{-1/2}^{1/2} |D_N (t)|dt;$$ it is immediate that $|T_N f|\le L_N \Vert f \Vert _\infty.$ $D_N(t)=\sum_{k=-N}^N e^{2\pi i kt}=\frac{\sin(\pi(2N+1)t)}{\sin(\pi t)}$ has a finite number of zeros so $sgn D_N(t)$ has a finite number of jump discontinuities.

Therefore, by modifying it on a small neighborhood of each discontinuity, we can form a continuous function $f$ such that $\Vert f\Vert_\infty =1$ and $|T_N f|\ge L_N -\epsilon.$ Hence, $\Vert T_N\Vert =L_N$.

I don't completely understand how the bolded sentence is true. I can see that by modifying $g:=sgn D_N$, say by linearly connecting near the discontinuous points, that we can find a continuous $f$ that approximates $D_N$ uniformly, but I can't guarantee that this function will be periodic, which is required to be on the space $X$. How can we construct this approximation? I would greatly appreciate any help.

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Hint: Think of the following simple situation: Let $f=1$ on $[0,1/2),$ $f=-1$ on $[1/2,1].$ For small $\epsilon>0,$ let $g_\epsilon$ be the continuous piecewise linear function whose graph connects the points $(0,1),(1/2-\epsilon,1),(1/2+\epsilon,-1),(1,-1).$ What is $\int_0^1|f-g_\epsilon|?$

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    That is the method I thought of, but then I need $g_\epsilon$ to be periodic as well as being continuous in this case, so how can I guarantee that it is periodic as well? – 2017-02-11
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    That's no problem. At the expense of two intervals of length $\epsilon$ at the endpoints, you can make $g_\epsilon =0$ at the endpoints. – 2017-02-11