Given nnn functions f1,f2,,fnf_1, f_2, cdots, f_nf1,f2,,fn, where fi={∣arctan∣π2f_i = left{ begin{aligned}. end{aligned} right.fi=∣arctan∣2π For each function fif_ifi, determine if there is an xix_ixi that j∈{1,2,,i1,i+1,,n},fi
Given nnn functions f1(x),f2(x),⋯ ,fn(x)f_1(x), f_2(x), cdots, f_n(x)f1(x),f2(x),⋯,fn(x), where fi(x)={∣arctan(kisec(x−ai))∣ (x≠ai+(k+12)π (k=0,±1,±2,⋯ ))π2 (x=ai+(k+12)π (k=0,±1,±2,⋯ ))f_i(x) = left{ begin{aligned}
|arctan(k_isec(x - a_i))| ; & ; (x neq a_i + (k+frac{1}{2})pi , (k = 0, pm 1, pm 2, cdots)) \
frac{pi}{2} ; & ; (x = a_i + (k+frac{1}{2})pi , (k = 0, pm 1, pm 2, cdots))
end{aligned} right.fi(x)=⎩⎪⎨⎪⎧∣arctan(kisec(x−ai))∣2π(x=ai+(k+21)π(k=0,±1,±2,⋯))(x=ai+(k+21)π(k=0,±1,±2,⋯)) For each function fi(x)f_i(x)fi(x), determine if there is an xix_ixi that ∀j∈{1,2,⋯ ,i−1,i+1,⋯ ,n},fi(xi)
