Continuous functions

## Continuous functions
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            ## Continuity at a point
            #### Definition
            A function $f:I\to\mathbb R$ is *continuous at* $a\in I$ if

$\displaystyle \lim\_{x\to a} f(x) = f(a)$.

Note: this requires the limit to exist.
            
            ---
            ## Continuity at a point
            #### Definition
            A function $f:I\to\mathbb R$ is *continuous at* $a\in I$ if

$\displaystyle \lim\_{x\to a} f(x) = f(a)$.

#### Definition
            A function $f:I\to\mathbb R$ is *left continuous continuous at* $a\in I$ if

#### Definition
            A function $f:I\to\mathbb R$ is *right continuous continuous at* $a\in I$ if
            
            ---
            ## Continuity at a point
            #### Definition
            A function $f:I\to\mathbb R$ is *continuous at* $a\in I$ if

$\displaystyle \lim\_{x\to a} f(x) = f(a)$.

#### Definition
            A function $f:I\to\mathbb R$ is *left continuous continuous at* $a\in I$ if

$\displaystyle \lim\_{x\to a^-} f(x) = f(a)$.

#### Definition
            A function $f:I\to\mathbb R$ is *right continuous continuous at* $a\in I$ if

$\displaystyle \lim\_{x\to a^+} f(x) = f(a)$.
            
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            ## Continuity at a point examples
            #### Example
            ![Desmos plot of y=x cubed minus x](gfx/x3-x.png "Desmos plot of y=x^3-x")

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            ## Continuity at a point examples
            #### Example
            ![Desmos plot of y=1 on (abs x plus 1)](gfx/1onabsxplus1.png "Desmos plot of y=1/(|x|+1)")

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            ## Continuity at a point examples
            #### Example
            ![Desmos plot of y=sin(x)](gfx/sinx.png "Desmos plot of y=sin(x)")

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            ## Continuity at a point examples
            #### Example
            ![Desmos plot of y=x plus the ceiling function of x](gfx/xplusceilxdisc.png "Desmos plot of y=x+ceil(x)")

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            ## Continuity on intervals
            #### Definition
            A function $f:I\to\mathbb R$ is *continuous at* $a\in I$ if
            $\displaystyle \lim\_{x\to a} f(x) = f(a)$.
            #### Definition
            A function $f:(a,b)\to\mathbb R$ is *continuous on* $(a,b)$ if
            $f$ is continuous at $y$ for all $y\in(a,b)$.
            
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            ## Continuity on intervals
            #### Definition
            A function $f:(a,b)\to\mathbb R$ is *continuous on* $(a,b)$ if
            $f$ is continuous at $y$ for all $y\in(a,b)$.
            #### Definition
            $f:[a,b]\to\mathbb R$ is *continuous on* $[a,b]$ if
            
            ---
            ## Continuity on intervals
            #### Definition
            A function $f:(a,b)\to\mathbb R$ is *continuous on* $(a,b)$ if
            $f$ is continuous at $y$ for all $y\in(a,b)$.
            #### Definition
            $f:[a,b]\to\mathbb R$ is *continuous on* $[a,b]$ if

$f$ is continuous on $(a,b)$ and $f$ is left continuous at $b$ and $f$ is right continuous at $a$.
            
            ---
            ## Continuity on intervals
            #### Definition
            A function $f:(a,b)\to\mathbb R$ is *continuous on* $(a,b)$ if
            $f$ is continuous at $y$ for all $y\in(a,b)$.
            #### Definition
            $f:[a,b]\to\mathbb R$ is *continuous on* $[a,b]$ if

$f$ is continuous on $(a,b)$ and $f$ is left continuous at $b$ and $f$ is right continuous at $a$.

#### Definition
            $f:(a,b]\to\mathbb R$ is *continuous on* $(a,b]$ if

#### Definition
            $f:[a,b)\to\mathbb R$ is *continuous on* $[a,b)$ if
            
            ---
            ## Continuity on intervals
            #### Definition
            A function $f:(a,b)\to\mathbb R$ is *continuous on* $(a,b)$ if
            $f$ is continuous at $y$ for all $y\in(a,b)$.
            #### Definition
            $f:[a,b]\to\mathbb R$ is *continuous on* $[a,b]$ if

$f$ is continuous on $(a,b)$ and $f$ is left continuous at $b$ and $f$ is right continuous at $a$.

#### Definition
            $f:(a,b]\to\mathbb R$ is *continuous on* $(a,b]$ if

$f$ is continuous on $(a,b)$ and $f$ is left continuous at $b$.

#### Definition
            $f:[a,b)\to\mathbb R$ is *continuous on* $[a,b)$ if

$f$ is continuous on $(a,b)$ and $f$ is right continuous at $a$.
            
            ---
            ## Continuity on intervals
            #### Definition
            A function $f:(a,b)\to\mathbb R$ is *continuous on* $(a,b)$ if
            $f$ is continuous at $y$ for all $y\in(a,b)$.
            #### Definition
            $f:[a,b]\to\mathbb R$ is *continuous on* $[a,b]$ if

$f$ is continuous on $(a,b)$ and $f$ is left continuous at $b$ and $f$ is right continuous at $a$.

#### Definition
            Suppose $X\subseteq\mathbb R$.
            Then $f:X\to\mathbb R$ is *continuous on* $X$ if

$X$ is a union of disconnected intervals and $f$ is continuous on each interval that makes up its domain.

By "disconnected intervals", we mean that the intervals cannot join up.
            
            ---
            ## Continuity on intervals
            #### Example
            ![Desmos plot of y=x cubed minus x](gfx/x3-x.png "Desmos plot of y=x^3-x")

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            ## Continuity on intervals
            #### Example
            ![Desmos plot of y=1 on (abs x plus 1)](gfx/1onabsxplus1.png "Desmos plot of y=1/(|x|+1)")

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            ## Continuity on intervals
            #### Example
            ![Desmos plot of y=sin(x)](gfx/sinx.png "Desmos plot of y=sin(x)")

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            ## Continuity on intervals
            #### Example
            ![Desmos plot of y=x plus the ceiling function of x](gfx/xplusceilxdisc.png "Desmos plot of y=x+ceil(x)")

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            ## Continuity on intervals
            #### Example
            ![Desmos plot of y=x plus the ceiling function of x](gfx/xplusceilxcts.png "Desmos plot of y=x+ceil(x)")