You can solve for $y$ to get the two branches. Note that the plot is wrong and seems to think they are left/right branches, but actually they are top/bottom:

{f1, f2} = y /. Solve[f[x, y] == 0, y] /. C[1] -> 0; Plot[{f1, f2}, {x, 3.45, 3.5}] 

These expressions are both ArcTan of a large inner expression. We can strip the ArcTan and subtract:

result=FullSimplify[f1[[1]] - f2[[1]]] 

... which gives:

-((Sqrt[2] Csch[\[Pi] x]^2 \[Sqrt](-x^2 (2 Cosh[(\[Pi] x)/3] + Cosh[\[Pi] x])^2 (155392 - 153856 x^2 + 8608 x^4 - 144 x^6 - x^8 - 256 (-6 + x) (6 + x) (32 - 28 x^2 + x^4) Cosh[( 2 \[Pi] x)/3] - 64 (-3776 + 2768 x^2 - 212 x^4 + 3 x^6) Cosh[(4 \[Pi] x)/ 3] - 64 (-12 + x^2) (208 - 96 x^2 + 3 x^4) Cosh[ 2 \[Pi] x] - 128 (-12 + x^2)^2 (-4 + x^2) Cosh[(8 \[Pi] x)/ 3] + (-12 + x^2)^4 Cosh[(10 \[Pi] x)/3]) Sinh[(\[Pi] x)/ 3]^8))/(16 - 24 x^2 + x^4 + (4 + x^2)^2 Cosh[(4 \[Pi] x)/3])) 

That means they touch at a point provided this equation has a root in the vicinity of $34/10<x<35/10$. Numerically we can maximize it, but this is not a proof:

NMaximize[{Re@result, 34/10 < x < 35/10}, x, WorkingPrecision -> 50] (* {0, {x -> 3.4657284157760663194818753686596237797063058094688}} *) 

However, we can take a subexpression from this equation and check:

Resolve[Exists[x, 34/10 < x < 35/10, (155392 - 153856 x^2 + 8608 x^4 - 144 x^6 - x^8 - 256 (-6 + x) (6 + x) (32 - 28 x^2 + x^4) Cosh[(2 \[Pi] x)/3] - 64 (-3776 + 2768 x^2 - 212 x^4 + 3 x^6) Cosh[(4 \[Pi] x)/3] - 64 (-12 + x^2) (208 - 96 x^2 + 3 x^4) Cosh[2 \[Pi] x] - 128 (-12 + x^2)^2 (-4 + x^2) Cosh[(8 \[Pi] x)/ 3] + (-12 + x^2)^4 Cosh[(10 \[Pi] x)/3]) == 0], Reals] (* result: True *) 

This 'proves' that a zero exists and that they do meet at a point. Though without knowing what Resolve/Exists does internally, it's not the most satisfying result.