op amp rail to rail voltage follower latching up
A couple of days ago I tested some opamp in voltage follower configuration. 
All op-amps were bought from a reputable vendor (TME).

 

The input signal was coming from the function generator, \$F = 1\textrm{kHz}\$ and the signal amplitude was \$10V\$ peak to peak. 

And all Opamp's was also supplied from symmetrical supply \$+/-5V\$. 

Hence we can easily examine the allowed input common mode voltage range.

The first one was \$\textrm{LM}358\$ from ST. 



As you can clipping accrue for \$Vin &gt; (Vcc-1.4V)\$

And no sign of a famous crossover distortion due to lack of load resistance.

But reducing the input signal amplitude and adding a load resistance reveals the crossover distortion. 

 

The second opamp was \$\textrm{TS922}\$ from ST



As you can see \$Vin = Vout\$ as expected form rail-to-rail input and output opamp.

No sign of "latching-up". 

Hence, if your setup is correct I vote for a counterfeit part from China. Not a big surprise either.  

Next one was \$\textrm{TL}072\$ from Ti.  



As you can see we got a clipping for \$Vin &gt; (Vcc-0.6V)\$ or so.
And when the input signal approaches \$V_{EE} - 1.4V\$ phase inversion phenomenon manifests itself. 

And finally the \$\textrm{NE}5532\$ from Philips.