One-sided derivatives are defined by one-sided limits. In this post, we will introduce the right-hand derivative and the left-hand derivative of a function at a given point.

#### One-sided limits

Let \(f\) be a real-valued function.

- Assume that the domain of \(f\) contains the open interval \((c,d)\). We say the right-hand limit of \(f\) at \(c\) is \(L\) if for each positive real number \(\epsilon\) there is a corresponding positive real number \(\delta\) such that \(c < x < c + \delta\) implies \(\vert f(x) – L \vert < \epsilon\). In such a case, we write $$\lim_{x \to c^{+}} f(x) = L.$$
- Assume that the domain of \(f\) contains the open interval \((b,c)\). We say the left-hand limit of \(f\) at \(c\) is \(L\) if for each positive real number \(\epsilon\) there is a corresponding positive real number \(\delta\) such that \(c-\delta < x < c\) implies \(\vert f(x) – L \vert < \epsilon\). In such a case, we write $$\lim_{x \to c^{-}} f(x) = L.$$

The reader should be able to easily prove the following result:

**Result**. Let \(f\) be a real-valued function, \(I\) an open interval with \(c\in I\) and \(L\) a real number. Assume that \(f\) is definable on \(I\) except perhaps at \(c\) itself. Then, the limit of \(f\) is \(L\) as \(x\) approaches to \(c\) if and only if the right-hand limit of \(f\) at \(c\) is \(L\) and the left-hand limit of \(f\) at \(c\) is \(L\). In other words, $$\lim_{x \to c} f(x) = L$$ if and only if $$\lim_{x \to c^{+}} f(x) = L \text{ and } \lim_{x \to c^{-}} f(x) = L.$$

**Exercise**. Compute the right-hand and the left-hand limit of \(f(x) = \displaystyle \frac{\sin x}{\vert x \vert}\) at \(0\). Deduce that the limit of \(f(x) = \displaystyle \frac{\sin x}{\vert x \vert}\) at \(0\) does not exist.

**Remark**. By one-sided limits, we mean either the right-hand or the left-hand limit.

#### One-sided derivatives

The right-hand derivative of a function \( f \) at a point \( c \) exists if $$\lim_{h \to 0^{+} } \frac{f(c+h) – f(c)}{h}$$ exists.

Similarly, the left-hand derivative of a function \(f\) at a point \(c\) exists if $$\lim_{h \to 0^{-}}\frac{f(c+h) – f(c)}{h}$$ exists.

**Remark**. By one-sided derivatives, we mean either the right-hand or the left-hand derivative.

**Exercise**. Find the one-sided derivatives of the function $$f(x) = \vert x -1 \vert + \vert x -2 \vert$$ at the points \(x=1\), \(x=2\) and \(x=3\).

**Exercise**. Find the one-sided derivatives of the floor function \(\lfloor x \rfloor\) defined by $$\lfloor x \rfloor=\max \{m\in\mathbb{Z}: m \leq x\},$$ at an arbitrary point \(x\) if it exists.