Symmetric Derivative - Examples

Examples

1. The modulus function,
For absolute value function, or the modulus function, we have, at ,

\begin{matrix} \\ f_s(0)= \lim_{h \to 0}\frac{f(0+h) - f(0-h)}{2h} \\ \\ f_s(0)= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\ \\ f_s(0)= \lim_{h \to 0}\frac{\left\vert h \right\vert - \left\vert -h \right\vert}{2h} \\ \\ f_s(0)= \lim_{h \to 0}\frac{h-(-(-h))}{2h} \\ \\ f_s(0)= 0 \\ \end{matrix}

only, where remember that and, and hence is equal to only! So, we observe that the symmetric derivative of the modulus function exists at ,and is equal to zero, even if its ordinary derivative won't exist at that point (due to a "sharp" turn in the curve at ).

2. The function
For the function, we have, at ,

\begin{matrix} \\ f_s(0)= \lim_{h \to 0}\frac{f(0+h) - f(0-h)}{2h} \\ \\ f_s(0)= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\ \\ f_s(0)= \lim_{h \to 0}\frac{1/h^2 - 1/(-h)^2}{2h} \\ \\ f_s(0)= \lim_{h \to 0}\frac{1/h^2-1/h^2}{2h} \\ \\ f_s(0)= 0 \\ \end{matrix}

only, where again, and . See that again, for this function, its symmetric derivative exists at, its ordinary derivative does not occur at, due to discontinuity in the curve at (i.e. essential discontinuity).

3. The Dirichlet function, defined as:

f(x) = \begin{cases} 1, & \text{if }x\text{ is rational} \\ 0, & \text{if }x\text{ is irrational} \end{cases}

may be analysed to realize that it has symmetric derivatives but not, i.e. symmetric derivative exists for rational numbers bur not for irrational numbers.

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