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I understand how Guitar works. Pitch/Frequency of sound produced depends on length of vibrating string. So we have different frets to product different notes. Fine.

However I am curious to understand how the fret itself is designed. When finger moved to new fret we hear new sound. But within the same fret, even if finger is in different position (so here length of vibrating string changes) we hear the same note. Why and how? How Frets are designed in such a way?

If it is just natural, then are music notes discrete? I believe classical musical like Indian Carnatic music assumes music notes are continuous. So I don't think music notes are discrete, instead Guitar frets are designed in such a way to make it sound discrete. Please advise. Thank you :)

Actually, this is asked using inaccurate facts.

The position of the fingertip within the space of two fretwires does produce a note - but the pitch of that note does vary considering where in that space the finger is. That's how we get vibrato - classical style. By rolling the fingertip towards and away from the bridge, within one fret space, the note changes, albeit slightly.

That aside, the premise is that wherever the string is pressed on a particular fret space, the note is the same. That's because the string will sound from bridge (or saddle) to that fretwire. It's always the same distance, so will always be the same note. Essentially.

There's also the fact that within a fret space, extra pressure can be applied. This will sharpen a fretted note. It's how one student of mine makes vibrato.

The essence of frets is that it splits the fingerboard into discrete diatonic parts, and well made fretboards will (basically) mean the same note gets played on the same fret on the same string every time - assuming the guitar's tuned properly!

There are fretless guitars - just like there are fretless bass guitars - but while they are far less easy to play in tune, due to inaccuracy of fingering as the fretwires aren't there to help - they behave more like the violin family, consequently are more able to play some note with more accuracy than those compromised by being fretted.

Actually music notes are not discrete. The guitar, like the piano, is designed to have equal tempered tuning rather than just tuning. In that tuning system notes are discrete, half step = 12th root of 2. Even in just tuning we only have 7 notes in the diatonic scale but we are free to make slight deviations and some cultures do use quarter steps (a half of a half step) in their scales.

As for where you place the finger, there is a "correct" way to do this. The finger should be placed just behind the fret. This is to avoid letting the string slide or slap against the fret as it vibrates. If you fret the note with your finger somewhere in the middle of the space between frets you are not really creating a boundary condition between the bridge and the fret. The string is not truly "fixed" at the fret. If you don't overdrive the string mechanically it wont matter but if you play with too much force the string will not stay fixed at the fret but slide along it, even lifting slightly and slapping down on the fret. This will cause buzzing and a sitar like sound. This is a bigger issue on the classical. When you have complex chords forms we sometimes have no choice and simply need to get a finger somewhere in the space but if we can, we should get them pinched right behind the fret. Along the same lines a common misconception is that the string needs to be pressed down to the wood of the finger board. This is also not true. Only enough pressure is required to stop buzzing against the fret and get a good tone. In theory one should be able to slide a piece of paper under the string even when fretted. This is as true for the electric as it is for the acoustic or classical but not as necessary. In my experience the electric is a little more forgiving (if set up well). If you do press the string down to the wood you will bend it our of tune. If you experiment, playing just one note varying all these different parameters you will hear it.

Why and how? How Frets are designed in such a way?

If you actually have a guitar in your hands, it's very obvious how frets work. The fret is raised above the fingerboard, so if you place your finger behind a fret, the string will become 'stopped' at that fret such that the 'speaking length' of the string is the length between the bridge and the fret. Here's a simplified diagram showing a guitar with one fret:

From your deleted answer:

I got to know now how it works. So Frets do have a divider between them. So whenever fingers press against a fret, whatever finger's position within the same fret, it is this divider which gets in contact with string so length of vibrating string doesn't change, except when fingers moved to new fret new divider will come into play.

You got the idea ok, but 'fret' is the name for the 'divider'.

If it is just natural, then are music notes discrete? I believe classical musical like Indian Carnatic music assumes music notes are continuous. So I don't think music notes are discrete, instead Guitar frets are designed in such a way to make it sound discrete.

That's quite a complicated question! In most contexts in western music, notes are assumed to have a single particular pitch, but it's also understood that they can be bent, or that effects like vibrato or glissando can be used to change the pitch. A fretted guitar can't easily do a glissando, but it can do bends and vibrato through the mechanisms described in other answers.

The pitch of the note (frequency of vibration) is set by (at least) 3 factors;

(1) the length of the vibrating portion of the string,

(2) the mass of the vibrating portion of the string, and

(3) the tension of the string.

So:

(1) The vibrating length of the string between fret and saddle remains the same, regardless of where you fret. The frets are intended to make the notes discrete.

(2) I don't know of any way to change the mass of the strings on the fly.

(3) Varying the tension, like by bending or pulling the string or bending the neck, will change the note.

Hmm, would the stiffness of the string material affect frequency as well?