Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,

Let $f$ be a periodic BV function with atleast one jump, and let $S^f_N(x)$ be its Fourier partial sum, $N = 1,2,3,...$ Let $t_N = TV(S^f_N(x))$ where $TV(f)$ denotes the total variation of the function $f$. Does $$\lim_{N\to\infty}\{t_N-TV(f)\}$$ exists and goes to zero or to $\infty$? I think it goes to $\infty$, but not sure. Request your answers.