We report the first measurement of $\Lambda$ and $\overline{\Lambda}$ hadron transverse polarization inside jets in unpolarized proton-proton collisions, which is attributed to the polarizing fragmentation function (PFF). 
The polarization was measured as a function of the jet transverse momentum ($p_T$), the fraction of the jet momentum ($z$) carried by $\Lambda$($\overline{\Lambda}$) hadrons, and the transverse momentum of $\Lambda(\overline{\Lambda})$ hadrons relative to the jet axis ($j_T$).
$\Lambda$ polarization shows a clear dependence on the jet $p_T$, changing sign from negative to positive from low to high jet $p_T$, while $\overline{\Lambda}$ polarization mostly remains negative. 
These data cover a wider range of jet energies than existing data in $e^+e^-$ annihilations, and may also provide the first constraints on the gluon PFF. %which is not accessible in $e^+e^-$ annihilation.  
These results provide crucial tests of the TMD evolution effect and its universality for PFF, and are thus important input to a complete understanding of the transverse polarization mechanism.