How to determine if the character can jump or not

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If you have a wall mesh, then you'll need to check the normals of the collision then. The collision normals (CollisionNX() etc etc) return basically an angle at which the collision occured. Using this, you could determine whether the geometry encountered is steep enough to be considered a wall.

If bbKeyHit(bbkey_x)
bbLinePick(bbEntityX(character),bbEntityY(character),bbEntityZ(character),0,0,1.2)
  If onground(character) = 1 Then charyspd = 0.5
    If bbPickedEntity()<>0
      charyspd = 0.5
    End If
End If


I reckon it didn't work, because your character is still on the ground. You can't wall jump whilst standing on the ground, and when you do jump, the onground() would surely equal 0, invalidating the if statement.