Sketch the graph of f(x)=x-[x] ; then find each of the...

Sketch the graph of $f(x)=x-[x] ;$ then find each of the following or state that it does not exist.
(a) $f(0)$
(b) $\lim _{x \rightarrow 0} f(x)$
(c) $\lim _{x \rightarrow 0^{-}} f(x)$
(d) $\lim _{x \rightarrow 1 / 2} f(x)$

Key Concepts

One-sided Limits

One-sided limits focus on the behavior of a function as the variable approaches a specific point from either the left or the right. They are essential in analyzing functions with discontinuities, determining if the function behaves differently from different directions, which is particularly important when the function's definition changes abruptly at certain points.

Graphical Analysis of Piecewise Functions

Graphical analysis involves sketching and understanding the shape and behavior of functions which are defined by different expressions on different intervals. For functions like the fractional part function, this analysis helps in visualizing the recurring patterns, discontinuities, and the overall structure of the function, which is instrumental in evaluating function values and limits.

Floor Function

The floor function, often denoted as [x], assigns to each real number the greatest integer less than or equal to that number. It creates a step-like behavior and introduces discontinuities at integer points, which are crucial for understanding how functions defined in relation to it behave, especially when considering limits and graph sketching.

Fractional Part Function

Defined as f(x) = x - [x], the fractional part function extracts the non-integer part of a real number, yielding values in the interval [0, 1). Its repetitive, periodic behavior on each interval [n, n+1), where n is an integer, provides a clear example of a piecewise defined function with jump discontinuities at the integers.

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