(sec:bra:real-valued-functions)= # Real Valued Functions ## Real Valued Functions ```{index} Real valued function ``` ```{prf:definition} Real valued function :label: def-bra-real-valued-function A (partial) *real valued function* is a function whose values are real numbers. Let $X$ be a set. Then $f : X \to \RR$ is a real valued function from $X$ to $\RR$. ``` ```{index} Set of real valued functions ``` ```{prf:definition} The set of real valued total functions :label: def-bra-rvf-set The set $\FFF (X, \RR)$ denotes the set of all real valued (total) functions from $X$ to $\RR$. ``` ```{prf:definition} The vector space of real valued functions :label: def-bra-real-valued-function-vector-space The set $\FFF (X, \RR)$ can be turned into a vector space over the field $\RR$ with the following operations. Let $f,g \in \FFF (X, \RR)$. Vector addition: $$ f + g : x \mapsto f(x) + g(x) \Forall x \in X. $$ Additive identity: $$ \bzero : x \mapsto 0 \text{ with } \Forall x \in X. $$ Scalar multiplication: $$ cf : x \mapsto c f(x) \Forall x \in X. $$ pointwise multiplication: $$ f g: x \mapsto f(x) g(x) \Forall x \in X. $$ ``` ```{prf:definition} An algebra for partial functions :label: def-bra-rvpf-algebra An algebraic structure can be provided to partial functions too. Let $f,g$ be (partial) real valued functions from $X$ to $\RR$. Vector addition: $$ f + g : x \mapsto f(x) + g(x) \text{ with } \dom f + g = \dom f \cap \dom g. $$ Additive identity: $$ \bzero : x \mapsto 0 \text{ with } \dom \bzero = X. $$ Scalar multiplication: $$ cf : x \mapsto c f(x) \text{ with } \dom cf = \dom f. $$ pointwise multiplication: $$ f g: x \mapsto f(x) g(x) \text{ with } \dom f g = \dom f \cap \dom g. $$ ``` However, there are certain limitations/odd behaviors with the structure. * If $f$ and $g$ are such that $\dom f \cap \dom g = \EmptySet$. Then $f + g$ is an empty function. * The function $f + (-f)$ is 0 over $\dom f$ but not defined over $X \setminus \dom f$. Thus, it is not equal to the $\bzero$ function. Thus, an additive inverse doesn't exist. * Scalar multiplication with $0$ leads to a function which is $0$ only over $\dom f$. It is not defined over $X \setminus \dom f$. ```{index} Real valued function; partial order ``` ```{prf:definition} Partial order on real valued (total) functions :label: def-bra-rv-func-partial-order Since $\RR$ is ordered, hence a partial order can be defined on $\FFF (X, \RR)$. We say that $$ f \preceq g \iff f(x) \leq g(x) \Forall x \in X. $$ ``` Partial order cannot be easily defined for partial functions as it is unclear how to compare $f(x)$ and $g(x)$ at $x \in \dom f \triangle \dom g$. One possible way is: $$ f \preceq g \iff \dom f = \dom g \text{ and } f(x) \leq g(x) \Forall x \in \dom f. $$ ```{index} Bounded function ``` ```{prf:definition} Bounded function :label: def-bra-bounded-function A real valued (total) function $f:X \to \RR$ is called *bounded* if there exists a number $M \geq 0$ (depending on $f$) such that $$ | f(x)| \leq M \Forall x \in X. $$ A function which is not bounded is called *unbounded*. $f$ is called *bounded from above* by $a \in \RR$ if: $$ f(x) \leq a \Forall x \in X. $$ $f$ is called *bounded from below* by $b \in \RR$ if: $$ b \leq f(x) \Forall x \in X. $$ ``` Boundedness of partial real valued functions (with $\dom f \subset X$) is not useful as partial functions are typically extended (see below) with $f(x)$ assigned to $\infty$ at $x \notin \dom f$. In other words, partial functions are treated as unbounded outside their domain. ```{prf:proposition} :label: res-bra-rv-func-bounded-charac A real valued function is bounded if and only if it is bounded from above as well as below. ``` ```{rubric} See also ``` 1. The set of bounded (total) functions can be turned into a metric space. See {prf:ref}`ex-ms-bounded-functions-metric-space`. ## Graph * For a function $f : \RR^n \to \RR$, its graph is a subset of $\RR^{n+1}$. * We say that a point $(x, f(x))$ in the graph of $f$ is above (resp. below) of another point $(y, f(y))$ if $f(x) \geq f(y)$ (resp. $f(x) \leq f(y)$). * A line segment connecting the two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ is called a *chord* of the graph of the function. ## Epigraph ```{index} Epigraph ``` ```{prf:definition} Epigraph :label: def-bra-epigraph The *epigraph* of a real valued function $f: X \to \RR$ is defined as: $$ \epi f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, f(x) \leq t \}. $$ ``` The epigraph lies above (and includes) the graph of a function. ```{index} Strict epigraph ``` ```{prf:definition} Strict epigraph :label: def-bra-strict-epigraph The *strict epigraph* of a real valued function $f: X \to \RR$ is defined as: $$ \epi_s f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, f(x) < t \}. $$ ``` The strict epigraph lies above the graph of a function. ```{prf:theorem} Epigraph of pointwise maximum of two functions :label: res-bra-epigraph-intersection Let $f, g : X \to \RR$ be two different real valued functions. Let $h : X \to \RR$ with $\dom h = \dom f \cap \dom g$ be defined as $$ h(x) = \max(f(x), g(x)) \Forall x \in \dom h $$ Then $$ \epi h = \epi f \cap \epi g. $$ ``` ```{prf:proof} We first show that $\epi h \subseteq \epi f \cap \epi g$. 1. Let $(x, t) \in \epi h$. 1. Then $x \in \dom h$ and $h(x) \leq t$. 1. Hence $x \in \dom f$, $x \in \dom g$, $f(x) \leq t$ and $g(x) \leq t$. 1. Hence $(x,t) \in \epi f$ and $(x, t) \in \epi g$. 1. Hence $(x, t) \in \epi f \cap \epi g$. For the converse, we show that $\epi f \cap \epi g \subseteq \epi h$. 1. Let $(x, t) \in \epi f \cap \epi g$. 1. Then $(x, t) \in \epi f$ and $(x, t) \in \epi g$. 1. Thus $x \in \dom f$, $f(x) \leq t$, $x \in \dom g$ and $g(x) \leq t$. 1. Thus $x \in \dom f \cap \dom g = \dom h$. 1. Also, $h(x) = \max(f(x), g(x)) \leq t$. 1. Hence $(x, t) \in \epi h$. ``` This result can be generalized for an arbitrary family of functions. ```{prf:theorem} Epigraph of pointwise maximum of a family of functions :label: res-bra-epigraph-intersection-family Let $\{ f_i : X \to \RR \}_{i \in I}$ be a family of real valued functions indexed by $I$. Let $h : X \to \RR$ with $\dom h = \bigcap_{i \in I} \dom f_i$ be defined as $$ h(x) = \max \{f_i(x) \ST i \in I \} \Forall x \in \dom h $$ Then $$ \epi h = \bigcap_{i \in I} \epi f_i. $$ ``` ```{prf:proof} We first show that $\epi h \subseteq \bigcap_{i \in I} \epi f_i$. 1. Let $(x, t) \in \epi h$. 1. Then $x \in \dom h$ and $h(x) \leq t$. 1. Hence $x \in \dom f_i$ and $f_i(x) \leq t$ for every $i \in I$. 1. Hence $(x,t) \in \epi f_i$ for every $i \in I$. 1. Hence $(x, t) \in \bigcap_{i \in I} \epi f_i$. The argument for the converse is similar and left as an exercise. ``` ## Sub-level Sets ```{index} Sublevel set ``` ```{prf:definition} Sub-level set :label: def-bra-sub-level-set For a real valued function $f: X \to \RR$, the sublevel set for some $\alpha \in \RR$, denoted by $\sublevel(f, \alpha)$, is defined as $$ \sublevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \leq \alpha \}. $$ ``` ## Contours or Level Sets ```{index} Contour ``` ```{prf:definition} Contour :label: def-bra-contour For a real valued function $f: X \to \RR$, the contour for some $\alpha \in \RR$, denoted by $\contour(f, \alpha)$, is defined as $$ \contour(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) = \alpha \}. $$ ``` ## Hypograph ```{index} Hypograph ``` ```{prf:definition} Hypograph :label: def-bra-hypograph The *hypograph* of a real valued function $f: X \to \RR$ is defined as: $$ \hypo f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, t \leq f(x) \}. $$ ``` The epigraph lies above (and includes) the graph of a function. ## Super-level Sets ```{index} Superlevel set ``` ```{prf:definition} Super-level set :label: def-bra-super-level-set For a real valued function $f: X \to \RR$, the super-level set for some $\alpha \in \RR$, denoted by $\superlevel(f, \alpha)$, is defined as $$ \superlevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \geq \alpha \}. $$ ``` ## Extended Real Valued Functions ```{index} Extended real valued function ``` ```{prf:definition} Extended real-valued function :label: def-bra-extended-real-valued-function A function over a set $X$ is called an *extended real-valued function* if it can take any real value as well as the infinity values $-\infty$ and $\infty$. The signature of such a function is $f : X \to \ERL$ where $\ERL = \RR \cup \{ -\infty, \infty \}$. We also write the codomain as $\ERL = [-\infty, \infty]$. ``` ```{index} Effective domain ``` ```{prf:definition} Effective domain of an extended real valued function :label: def-bra-extension-domain For an extended valued function $\tilde{f} : X \to \ERL$, its *effective domain* is defined as: $$ \dom \tilde{f} \triangleq \{ x \in X \ST \tilde{f}(x) < \infty \}. $$ ``` ```{index} Extended real valued function; graph ``` ```{index} Extended real valued function; epigraph ``` ```{index} Extended real valued function; strict epigraph ``` ```{index} Extended real valued function; sublevel set ``` ```{index} Extended real valued function; contour ``` ```{index} Extended real valued function; hypograph ``` ```{index} Extended real valued function; superlevel set ``` ```{prf:definition} Graphs and level sets :label: def-bra-extended-value-func-graphs The epigraph, hypograph, sublevel, superlevel and contour sets of an extended valued function are defined in an identical manner. However, the graph is defined slightly differently. $$ & \graph f \triangleq \{ (x,t) \in X \times \ERL \ST x \in \dom f, f(x) = t \};\\ & \epi f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, f(x) \leq t \};\\ & \epi_s f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, f(x) < t \};\\ & \sublevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \leq \alpha \}; \\ & \contour(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) = \alpha \};\\ & \hypo f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, t \leq f(x) \};\\ & \superlevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \geq \alpha \}. $$ ``` ```{div} For an extended valued function, it is not necessary that $\graph f \subseteq \epi f$. 1. If $f(x) = \infty$, then $(x, \infty) \in \graph f$. However, $(x, \infty) \notin \epi f$. At the same time $(x, t) \notin \epi f$ for every $\RR$. 1. If $f(x) = -\infty$ then $(x, -\infty) \in \graph f$. However $(x, -\infty) \notin \epi f$. But $(x, t) \in \epi f$ for every $\RR$. ``` ```{index} Extended value extension ``` ```{prf:definition} Extended-value extension :label: def-bra-extended-value-extension Let $f: X \to \RR$ be a real valued (partial) function. We define its *extended-value extension* $\tilde{f} : X \to \ERL$ as $$ \tilde{f}(x) \triangleq \begin{cases} f(x) & \text{for} & x \in \dom f \\ \infty & \text{for} & x \notin \dom f \end{cases} $$ ``` The extension is pretty useful in analysis and optimization as it extends the domain to the whole of $X$. ```{index} Indicator function ``` ```{prf:definition} Indicator functions :label: def-bra-indicator-function Let $C$ be a subset of $X$. We define the *indicator function* for $C$ as: $$ I_C(x) = 0 \Forall x \in C. $$ By definition: $\dom I_C = C$. We can create an extended value extension of $I_C$ as: $$ \tilde{I}_C(x) \triangleq \begin{cases} 0 & x \in C \\ \infty & x \notin C. \end{cases} $$ ```