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"markdown": "# Continuity\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\n```\n\n\n\n\n---\n\n\nThe definition Google finds for *continuous* is *forming an unbroken whole; without interruption*.\n\n\nThe concept in calculus, as transferred to functions, is similar. Roughly speaking, a continuous function is one whose graph could be drawn without having to lift (or interrupt) the pencil drawing it.\n\n\nConsider these two graphs:\n\n::: {.cell hold='true' execution_count=4}\n\n::: {.cell-output .cell-output-display execution_count=5}\n{}\n:::\n:::\n\n\nand\n\n::: {.cell hold='true' execution_count=5}\n\n::: {.cell-output .cell-output-display execution_count=6}\n{}\n:::\n:::\n\n\nThough similar at some level - they agree at nearly every value of $x$ - the first has a \"jump\" from $-1$ to $1$ instead of the transition in the second one. The first is not continuous at $0$ - a break is needed to draw it - where as the second is continuous.\n\n\nA formal definition of continuity was a bit harder to come about. At [first](http://en.wikipedia.org/wiki/Intermediate_value_theorem) the concept was that for any $y$ between any two values in the range for $f(x)$, the function should take on the value $y$ for some $x$. Clearly this could distinguish the two graphs above, as one takes no values in $(-1,1)$, whereas the other - the continuous one - takes on all values in that range.\n\n\nHowever, [Cauchy](http://en.wikipedia.org/wiki/Cours_d%27Analyse) defined continuity by $f(x + \\alpha) - f(x)$ being small whenever $\\alpha$ was small. This basically rules out \"jumps\" and proves more useful as a tool to describe continuity.\n\n\nThe [modern](http://en.wikipedia.org/wiki/Continuous_function#History) definition simply pushes the details to the definition of the limit:\n\n\n> A function $f(x)$ is continuous at $x=c$ if $\\lim_{x \\rightarrow c}f(x) = f(c)$.\n\n\n\nThis says three things\n\n\n * The limit exists at $c$.\n * The function is defined at $c$ ($c$ is in the domain).\n * The value of the limit is the same as $f(c)$.\n\n\nThis speaks to continuity at a point, we can extend this to continuity over an interval $(a,b)$ by saying:\n\n\n> A function $f(x)$ is continuous over $(a,b)$ if at each point $c$ with $a < c < b$, $f(x)$ is continuous at $c$.\n\n\n\nFinally, as with limits, it can be convenient to speak of *right* continuity and *left* continuity at a point, where the limit in the defintion is replaced by a right or left limit, as appropriate.\n\n\n:::{.callout-warning}\n## Warning\nThe limit in the definition of continuity is the basic limit and not an extended sense where infinities are accounted for.\n\n:::\n\n##### Examples of continuity\n\n\nMost familiar functions are continuous everywhere.\n\n\n * For example, a monomial function $f(x) = ax^n$ for non-negative, integer $n$ will be continuous. This is because the limit exists everywhere, the domain of $f$ is all $x$ and there are no jumps.\n * Similarly, the basic trigonometric functions $\\sin(x)$, $\\cos(x)$ are continuous everywhere.\n * So are the exponential functions $f(x) = a^x, a > 0$.\n * The hyperbolic sine ($(e^x - e^{-x})/2$) and cosine ($(e^x + e^{-x})/2$) are, as $e^x$ is.\n * The hyperbolic tangent is, as $\\cosh(x) > 0$ for all $x$.\n\n\nSome familiar functions are *mostly* continuous but not everywhere.\n\n\n * For example, $f(x) = \\sqrt{x}$ is continuous on $(0,\\infty)$ and right continuous at $0$, but it is not defined for negative $x$, so can't possibly be continuous there.\n * Similarly, $f(x) = \\log(x)$ is continuous on $(0,\\infty)$, but it is not defined at $x=0$, so is not right continuous at $0$.\n * The tangent function $\\tan(x) = \\sin(x)/\\cos(x)$ is continuous everywhere *except* the points $x$ with $\\cos(x) = 0$ ($\\pi/2 + k\\pi, k$ an integer).\n * The hyperbolic co-tangent is not continuous at $x=0$ – when $\\sinh$ is $0$,\n * The semicircle $f(x) = \\sqrt{1 - x^2}$ is *continuous* on $(-1, 1)$. It is not continuous at $-1$ and $1$, though it is right continuous at $-1$ and left continuous at $1$.\n\n\n##### Examples of discontinuity\n\n\nThere are various reasons why a function may not be continuous.\n\n\n * The function $f(x) = \\sin(x)/x$ has a limit at $0$ but is not defined at $0$, so is not continuous at $0$. The function can be redefined to make it continuous.\n * The function $f(x) = 1/x$ is continuous everywhere *except* $x=0$ where *no* limit exists.\n * A rational function $f(x) = p(x)/q(x)$ will be continuous everywhere except where $q(x)=0$. (The function $f$ may still have a limit where $q$ is $0$, should factors cancel, but $f$ won't be defined at such values.)\n * The function\n\n\n\n$$\nf(x) = \\begin{cases}\n -1 & x < 0 \\\\\n 0 & x = 0 \\\\\n 1 & x > 0\n\\end{cases}\n$$\n\n\nis implemented by `Julia`'s `sign` function. It has a value at $0$, but no limit at $0$, so is not continuous at $0$. Furthermore, the left and right limits exist at $0$ but are not equal to $f(0)$ so the function is not left or right continuous at $0$. It is continuous everywhere except at $x=0$.\n\n\n * Similarly, the function defined by this graph\n\n::: {.cell hold='true' execution_count=6}\n\n::: {.cell-output .cell-output-display execution_count=7}\n{}\n:::\n:::\n\n\nis not continuous at $x=0$. It has a limit of $0$ at $0$, a function value $f(0) =1/2$, but the limit and the function value are not equal.\n\n\n * The `floor` function, which rounds down to the nearest integer, is also not continuous at the integers, but is right continuous at the integers, as, for example, $\\lim_{x \\rightarrow 0+} f(x) = f(0)$. This graph emphasizes the right continuity by placing a point for the value of the function when there is a jump:\n\n::: {.cell hold='true' execution_count=7}\n\n::: {.cell-output .cell-output-display execution_count=8}\n{}\n:::\n:::\n\n\n * The function $f(x) = 1/x^2$ is not continuous at $x=0$: $f(x)$ is not defined at $x=0$ and $f(x)$ has no limit at $x=0$ (in the usual sense).\n * On the Wikipedia page for [continuity](https://en.wikipedia.org/wiki/Continuous_function) the example of Dirichlet's function is given:\n\n\n\n$$\nf(x) =\n\\begin{cases}\n0 & \\text{if } x \\text{ is irrational,}\\\\\n1 & \\text{if } x \\text{ is rational.}\n\\end{cases}\n$$\n\n\nThe limit for any $c$ is discontinuous, as any interval about $c$ will contain *both* rational and irrational numbers so the function will not take values in a small neighborhood around any potential $L$.\n\n\n##### Example\n\n\nLet a function be defined by cases:\n\n\n\n$$\nf(x) = \\begin{cases}\n3x^2 + c & x \\geq 0,\\\\\n2x-3 & x < 0.\n\\end{cases}\n$$\n\n\nWhat value of $c$ will make $f(x)$ a continuous function?\n\n\nWe note that for $x < 0$ and for $x > 0$ the function is a simple polynomial, so is continuous. At $x=0$ to be continuous we need a limit to exists and be equal to $f(0)$, which is $c$. A limit exists if the left and right limits are equal. This means we need to solve for $c$ to make the left and right limits equal. We do this next with a bit of overkill in this case:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\n@syms x c\nex1 = 3x^2 + c\nex2 = 2x-3\ndel = limit(ex1, x=>0, dir=\"+\") - limit(ex2, x=>0, dir=\"-\")\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```{=html}\n \n\\[\nc + 3\n\\]\n\n```\n:::\n:::\n\n\nWe need to solve for $c$ to make `del` zero:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\nsolve(del, c)\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```\n1-element Vector{Sym}:\n -3\n```\n:::\n:::\n\n\nThis gives the value of $c$.\n\n\n## Rules for continuity\n\n\nAs we've seen, functions can be combined in several ways. How do these relate with continuity?\n\n\nSuppose $f(x)$ and $g(x)$ are both continuous on $I$. Then\n\n\n * The function $h(x) = a f(x) + b g(x)$ is continuous on $I$ for any real numbers $a$ and $b$;\n * The function $h(x) = f(x) \\cdot g(x)$ is continuous on $I$; and\n * The function $h(x) = f(x) / g(x)$ is continuous at all points $c$ in $I$ **where** $g(c) \\neq 0$.\n * The function $h(x) = f(g(x))$ is continuous at $x=c$ *if* $g(x)$ is continuous at $c$ *and* $f(x)$ is continuous at $g(c)$.\n\n\nSo, continuity is preserved for all of the basic operations except when dividing by $0$.\n\n\n##### Examples\n\n\n * Since a monomial $f(x) = ax^n$ ($n$ a non-negative integer) is continuous, by the first rule, any polynomial will be continuous.\n * Since both $f(x) = e^x$ and $g(x)=\\sin(x)$ are continuous everywhere, so will be $h(x) = e^x \\cdot \\sin(x)$.\n * Since $f(x) = e^x$ is continuous everywhere and $g(x) = -x$ is continuous everywhere, the composition $h(x) = e^{-x}$ will be continuous everywhere.\n * Since $f(x) = x$ is continuous everywhere, the function $h(x) = 1/x$ - a ratio of continuous functions - will be continuous everywhere *except* possibly at $x=0$ (where it is not continuous).\n * The function $h(x) = e^{x\\log(x)}$ will be continuous on $(0,\\infty)$, the same domain that $g(x) = x\\log(x)$ is continuous. This function (also written as $x^x$) has a right limit at $0$ (of $1$), but is not right continuous, as $h(0)$ is not defined.\n\n\n## Questions\n\n\n###### Question\n\n\nLet $f(x) = \\sin(x)$ and $g(x) = \\cos(x)$. Which of these is not continuous everywhere?\n\n\n\n$$\nf+g,~ f-g,~ f\\cdot g,~ f\\circ g,~ f/g\n$$\n\n::: {.cell hold='true' execution_count=10}\n\n::: {.cell-output .cell-output-display execution_count=11}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nLet $f(x) = \\sin(x)$, $g(x) = \\sqrt{x}$.\n\n\nWhen will $f\\circ g$ be continuous?\n\n::: {.cell hold='true' execution_count=11}\n\n::: {.cell-output .cell-output-display execution_count=12}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\nWhen will $g \\circ f$ be continuous?\n\n::: {.cell hold='true' execution_count=12}\n\n::: {.cell-output .cell-output-display execution_count=13}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nThe composition $f\\circ g$ will be continuous everywhere provided:\n\n::: {.cell hold='true' execution_count=13}\n\n::: {.cell-output .cell-output-display execution_count=14}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nAt which values is $f(x) = 1/\\sqrt{x-2}$ not continuous?\n\n::: {.cell hold='true' execution_count=14}\n\n::: {.cell-output .cell-output-display execution_count=15}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nA value $x=c$ is a *removable singularity* for $f(x)$ if $f(x)$ is not continuous at $c$ but will be if $f(c)$ is redefined to be $\\lim_{x \\rightarrow c} f(x)$.\n\n\nThe function $f(x) = (x^2 - 4)/(x-2)$ has a removable singularity at $x=2$. What value would we redefine $f(2)$ to be, to make $f$ a continuous function?\n\n::: {.cell hold='true' execution_count=15}\n\n::: {.cell-output .cell-output-display execution_count=16}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nThe highly oscillatory function\n\n\n\n$$\nf(x) = x^2 (\\cos(1/x) - 1)\n$$\n\n\nhas a removable singularity at $x=0$. What value would we redefine $f(0)$ to be, to make $f$ a continuous function?\n\n::: {.cell hold='true' execution_count=16}\n\n::: {.cell-output .cell-output-display execution_count=17}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nLet $f(x)$ be defined by\n\n\n\n$$\nf(x) = \\begin{cases}\nc + \\sin(2x - \\pi/2) & x > 0\\\\\n3x - 4 & x \\leq 0.\n\\end{cases}\n$$\n\n\nWhat value of $c$ will make $f(x)$ continuous?\n\n::: {.cell hold='true' execution_count=17}\n\n::: {.cell-output .cell-output-display execution_count=18}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nSuppose $f(x)$, $g(x)$, and $h(x)$ are continuous functions on $(a,b)$. If $a < c < b$, are you sure that $lim_{x \\rightarrow c} f(g(x))$ is $f(g(c))$?\n\n::: {.cell hold='true' execution_count=18}\n\n::: {.cell-output .cell-output-display execution_count=19}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nConsider the function $f(x)$ given by the following graph\n\n::: {.cell hold='true' execution_count=19}\n\n::: {.cell-output .cell-output-display execution_count=20}\n{}\n:::\n:::\n\n\nThe function $f(x)$ is continuous at $x=1$?\n\n::: {.cell hold='true' execution_count=20}\n\n::: {.cell-output .cell-output-display execution_count=21}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\nThe function $f(x)$ is continuous at $x=2$?\n\n::: {.cell hold='true' execution_count=21}\n\n::: {.cell-output .cell-output-display execution_count=22}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\nThe function $f(x)$ is right continuous at $x=3$?\n\n::: {.cell hold='true' execution_count=22}\n\n::: {.cell-output .cell-output-display execution_count=23}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\nThe function $f(x)$ is left continuous at $x=4$?\n\n::: {.cell hold='true' execution_count=23}\n\n::: {.cell-output .cell-output-display execution_count=24}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nLet $f(x)$ and $g(x)$ be continuous functions whose graph of $[0,1]$ is given by:\n\n::: {.cell hold='true' execution_count=24}\n\n::: {.cell-output .cell-output-display execution_count=25}\n{}\n:::\n:::\n\n\nWhat is $\\lim_{x \\rightarrow 0.25} f(g(x))$?\n\n::: {.cell hold='true' execution_count=25}\n\n::: {.cell-output .cell-output-display execution_count=26}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\nWhat is $\\lim{x \\rightarrow 0.25} g(f(x))$?\n\n::: {.cell hold='true' execution_count=26}\n\n::: {.cell-output .cell-output-display execution_count=27}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\nWhat is $\\lim_{x \\rightarrow 0.5} f(g(x))$?\n\n::: {.cell hold='true' execution_count=27}\n\n::: {.cell-output .cell-output-display execution_count=28}\n```{=html}\n\n\n\n```\n:::\n:::\n\n\n",
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