#math/calculus
A function $f : \mathbb{R}\to \mathbb{R}$ is continuous at point $c$, if for every $\varepsilon > 0$, there exists a $\delta >0$ such that $|f(c) - f(x)| < \varepsilon$ for all $x \in{} (c - \delta, c + \delta)$.
For a given point $c$ take some $y$ delta, call it $\varepsilon$. Can you find a $\delta$ such that $f(c) - min(f(c\pm \delta))$ and $max(f(c\pm \delta)) - f(c)$ are both less than $\varepsilon$?
Imagine first that there is step raise discontinuity. That means at some point $d$, we have an instantaneous positive gap on the $y-\text{axis}$ at $f(d)$. Note $f(d)$ cannot be both the low and high point, it must be one or the other, lets assume $f(d)$ is the high point. If we select an $\varepsilon$ that is smaller than the size of the gap, how can you select a $\delta$ such that it fits in the gap?
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% axes
\draw[->] (-0.5,0) -- (3.5,0) node[right]{$x$};
\draw[->] (0,-0.3) -- (0,3.5) node[above]{$y$};
% epsilon band
\fill[green!10] (-0.5,1.5) rectangle (3.5,2.5);
\draw[green!60!black, dashed] (-0.5,2.5) -- (3.5,2.5) node[right, font=\small]{$f(c)+\varepsilon$};
\draw[green!60!black, dashed] (-0.5,1.5) -- (3.5,1.5) node[right, font=\small]{$f(c)-\varepsilon$};
% delta band
\fill[blue!10] (1.3,0) rectangle (2.7,3.3);
\draw[blue!60, dashed] (1.3,0) -- (1.3,3.3);
\draw[blue!60, dashed] (2.7,0) -- (2.7,3.3);
\node[below, blue!70!black, font=\small] at (1.3,0){$c-\delta$};
\node[below, blue!70!black, font=\small] at (2.7,0){$c+\delta$};
% step function
\draw[purple, thick] (-0.5,1) -- (2,1);
\draw[purple, thick] (2,2) -- (3.5,2);
% open and closed circles
\draw[purple, thick] (2,1) circle (1.5pt);
\fill[purple] (2,2) circle (1.5pt);
% point label
\node[above right, font=\small] at (2,2){$(c, f(c))$};
% arrow showing escape
\draw[->, red, thick] (1.6,1) -- (1.6,1.45);
\node[red, font=\small, left] at (1.6,1.2){escapes!};
% c label
\node[below, font=\small] at (2,0){$c$};
\draw (2,0.05) -- (2,-0.05);
\end{tikzpicture}
\end{document}
```
You cannot find any $\delta$ where the distance from $f(d)$ is smaller than this $\varepsilon$ gap.
If the curve is continuous, you can just squeeze the $\delta$ closer to $c$ until the min and max distances to the point is
lt; \varepsilon$
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% axes
\draw[->] (-0.5,0) -- (3.5,0) node[right]{$x$};
\draw[->] (0,-0.3) -- (0,3.5) node[above]{$y$};
% epsilon band around f(c) where c=1.5, f(x)=0.5x^2, f(1.5)=1.125
\fill[green!10] (-0.5,0.725) rectangle (3.5,1.525);
\draw[green!60!black, dashed] (-0.5,1.525) -- (3.5,1.525) node[right, font=\small]{$f(c)+\varepsilon$};
\draw[green!60!black, dashed] (-0.5,0.725) -- (3.5,0.725) node[right, font=\small]{$f(c)-\varepsilon$};
% delta band (shrunk to fit: delta=0.2, so x in [1.3, 1.7])
\fill[blue!10] (1.3,0) rectangle (1.7,3.3);
\draw[blue!60, dashed] (1.3,0) -- (1.3,3.3);
\draw[blue!60, dashed] (1.7,0) -- (1.7,3.3);
\node[below, blue!70!black, font=\small] at (1.3,0){$c-\delta$};
\node[below, blue!70!black, font=\small] at (1.7,0){$c+\delta$};
% polynomial curve f(x) = 0.5x^2
\draw[purple, thick, domain=-0.3:2.6, samples=80] plot (\x, {0.5*\x*\x});
% point at c
\fill[orange!80!red] (1.5,1.125) circle (1.5pt);
\node[above right, font=\small] at (1.5,1.125){$(c, f(c))$};
% c label
\node[below, font=\small] at (1.5,0){$c$};
\draw (1.5,0.05) -- (1.5,-0.05);
% checkmark
\node[green!50!black, font=\small] at (1.5,0.4){curve stays inside!};
\end{tikzpicture}
\end{document}
```
## Differentiable Functions
Now think of any [[Differentiable Function]], if you kept zooming in forever, wouldn't you find a point at which the line basically looks like the tangent? Then finding a $\delta$ for any $\varepsilon$ would be cake.
## Continuous but not Differentiable
Take $f(x) = |x|$ as an example.
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% axes
\draw[->] (-2.5,0) -- (2.5,0) node[right]{$x$};
\draw[->] (0,-0.3) -- (0,2.8) node[above]{$y$};
% epsilon band around f(0)=0
\fill[green!10] (-2.5,0) rectangle (2.5,0.8);
\draw[green!60!black, dashed] (-2.5,0.8) -- (2.5,0.8) node[right, font=\small]{$f(c)+\varepsilon$};
\draw[green!60!black, dashed] (-2.5,0) -- (2.5,0);
% delta band clipped to function height (|x| at x=0.5 is 0.5)
\fill[blue!10] (-0.5,0) -- (-0.5,0.5) -- (0,0) -- (0.5,0.5) -- (0.5,0) -- cycle;
\draw[blue!60, dashed] (-0.5,0) -- (-0.5,0.5);
\draw[blue!60, dashed] (0.5,0) -- (0.5,0.5);
\node[below, blue!70!black, font=\small] at (-0.5,0){$c-\delta$};
\node[below, blue!70!black, font=\small] at (0.5,0){$c+\delta$};
% f(x) = |x|
\draw[purple, thick] (-2.2,2.2) -- (0,0) -- (2.2,2.2);
% corner point
\fill[orange!80!red] (0,0) circle (1.5pt);
\node[above right, font=\small] at (0.05,0.05){$(c, f(c))$};
% label
\node[above left, font=\small, purple] at (-1.5,2.3){$f(x) = |x|$};
% note
\node[green!50!black, font=\small] at (0,1.2){curve stays inside!};
% tick marks
\foreach \x in {-2,-1,1,2} {
\draw (\x,0.05) -- (\x,-0.05) node[below, font=\tiny]{\x};
}
\foreach \y in {1,2} {
\draw (0.05,\y) -- (-0.05,\y) node[left, font=\tiny]{\y};
}
% c label
\node[below, font=\small] at (0,-0.15){$c$};
\end{tikzpicture}
\end{document}
```