Probability density and mass functions

What is the maximum value a probability density function and a probability mass function can have? Also, if we take a pdf of a continuous distribution, will the pdf always be continuous? I am a little confused between the physical nature of these functions. I would really appreciate if someone could give an intuitive explanation for these, and how they relate to each other.

• Since probability can't be greater than 1 (an event can have at most a 100% chance of happening), the maximum value either of those functions could possibly take is 1. – Dave Oct 22 '16 at 12:05
• @Dave for pmf yes, for pdf no. – user251257 Oct 22 '16 at 12:59
• Pdf is non-negative, measurable, and integrates to $1$. Nothing more, nothing less. – user251257 Oct 22 '16 at 13:03
• @user251257 what if the pdf was equal to 1 between 0 and 1, and 0 everywhere else? – Dave Oct 22 '16 at 13:04
• @Dave the maximum of a pdf could be any positive number. I don't get your comment. – user251257 Oct 22 '16 at 13:05