𝙋𝙄𝙀𝘾𝙀𝙒𝙄𝙎𝙀 𝙁𝙐𝙉𝘾𝙏𝙄𝙊𝙉𝙎! 𝙋𝙇𝙀𝘼𝙎𝙀 𝙃𝙀𝙇𝙋!!!

Consider the function

where $a$ is some number. 

What is the largest value of $a$ such that the graph of $y=f(x)$ intersects every horizontal line at least once?

Thank you and please help! :)