Smooth Function

1. Smooth Function

1.1. Definition

We call a function \(f\) smooth if all of its derivatives exist and are continuous.

In mathematical notation: \[f \in C^\infty \quad \Longleftrightarrow \quad f^{(n)} \text{ exists and is continuous for every } n \in \mathbb{N}.\]

1.2. Examples

1.2.1. Smooth functions

  1. Exponential function

    \(f(x) = e^{x}\)

    All derivatives: \[f'(x) = e^{x}, \quad f''(x) = e^{x}, \quad f'''(x) = e^{x}, \quad \dots\] -> All exist and are continuous -> smooth

    \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = \(f(x)\), ymin = -1, ymax = 8, xmin = -2, xmax = 2, samples = 100, domain = -2:2, thick ] \addplot[blue, very thick] {exp(x)}; \end{axis} \end{tikzpicture}
  2. Sine function

    Derivatives: \[g'(x) = \cos x, \quad g''(x) = -\sin x, \quad g'''(x) = -\cos x, \quad \dots\] -> All continuous -> smooth

    \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = \(g(x)\), ymin = -1.5, ymax = 1.5, xmin = -6.5, xmax = 6.5, samples = 200, domain = -2*pi:2*pi, thick ] \addplot[red, very thick] {sin(deg(x))}; \end{axis} \end{tikzpicture}

1.2.2. Non-smooth function

  1. Absolute value

    h(x) = |x|

    Not differentiable at \(x = 0\) -> not smooth

    \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = \(h(x)\), ymin = -0.5, ymax = 3, xmin = -3, xmax = 3, samples = 100, domain = -3:3, thick ] \addplot[green!60!black, very thick] {abs(x)}; \end{axis} \end{tikzpicture} \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = \(h(x)\), ymin = -0.5, ymax = 3, xmin = -3, xmax = 3, samples = 100, domain = -3:3, thick ] \addplot[green!60!black, very thick] {abs(x)}; \end{axis} \end{tikzpicture}