
Reviewed for patch 7.3b

Favorable matchup
Reviewed for patch 7.3b
Most of Jax's damage comes from chaining attacks: Relentless Assault speeds up every hit and Grandmaster's Might rewards every third attack in a row. Ground Slam lowers his attack speed right when he needs it, and Malphite's passive armor softens the rest.
Relentless Assault raises Jax's attack speed with consecutive hits, and Grandmaster's Might's passive adds magic damage on every third attack in a row. Ground Slam cuts his attack speed for a few seconds: every hit comes later and his passive window gets harder to complete.
Counter Strike dodges basic attacks, not abilities. Seismic Shard and Ground Slam still land during his stance, though for somewhat less damage. What you shouldn't do is activate Thunderclap and throw empowered basic attacks into that defense.
Thunderclap's passive gives Malphite armor, which reduces Jax's basic attacks and Leap Strike. It isn't immunity: Empower and his ultimate's passive deal magic damage, so don't drag out a duel believing armor absorbs everything.
Answer with Ground Slam when Jax starts chaining attacks, not before he commits. If he activates Counter Strike, leave the radius of the closing stun and come back with Seismic Shard when it ends.
From Jax's side, don't start the attack chain while Malphite still has Ground Slam. Bait it out with a short leap and Counter Strike, step away, and come back while it's on cooldown. Only buy magic resist if his ability damage is actually punishing you; your item scaling is real, but not at the pace he sets in lane.

Favorable matchup
Reviewed for patch 7.3b
Blinding Dart makes Jax's attacks deal no damage while he's blinded, and almost his entire kit is empowered attacks. Teemo also punishes every step toward the wave from range. His risk is Counter Strike: during that stance your basic attacks get dodged.
Blinding Dart makes the target's attacks deal no damage. Jax relies on them: Empower charges his next attack, Relentless Assault builds hit by hit, and Grandmaster's Might rewards every third attack. Throw it when he jumps on you, not to poke.
Counter Strike dodges every attack and ends with a stun around him. Your Toxic Shot basic attacks won't land during the stance. Stop attacking, use Move Quick to leave the radius, and resume once the stun has gone off.
Poke every time Jax walks up to a minion without his leap available. If he leaps and activates Counter Strike, getting out of the radius comes first; save the blind for when he starts attacking, not the first contact.
For Jax, this matchup is won by waiting out the blind. Fake the engage with a short Leap Strike, draw out Blinding Dart, and go back in while it's on cooldown. Counter Strike protects you from his attacks and forces him away: use it before you get blinded, not after. Clear mushrooms on your retreat paths before you commit to a chase.

Execution matters
Reviewed for patch 7.3b
Jayce pokes Jax with Shock Blast, an ability Counter Strike doesn't dodge, and can push him away with Thundering Blow before the closing stun reaches him. It's situational: it demands mana and good stance choices, because an itemized Jax wins a long duel.
Shock Blast is an ability, so Counter Strike doesn't dodge it. Fired through Acceleration Gate it travels further and faster, letting you poke from outside Leap Strike range.
Thundering Blow knocks the target back. Counter Strike's stun hits nearby enemies when the stance ends: if you push him away while it's active, you can end up outside that radius and cut the trade short.
Poke in cannon stance while Jax can't leap onto you. Only swap to hammer when you need Thundering Blow to push him off, or to finish a weakened Jax with a clear way out.
Playing Jax, approach from behind your minions so Shock Blast explodes before it reaches you. When Jayce spends Thundering Blow, go in with Leap Strike and Counter Strike: without that knockback he has no way out of the stun. You scale better in long duels, so don't give up health chasing him in the early levels.