2.5. Real Valued Functions#

2.5.1. Real Valued Functions#

Definition 2.45 (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 \(\RR\).

Definition 2.46 (The set of real valued total functions)

The set \(\FFF (X, \RR)\) denotes the set of all real valued (total) functions from \(X\) to \(\RR\).

Definition 2.47 (The vector space of real valued functions)

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:

\[ f g: x \mapsto f(x) g(x) \Forall x \in X. \]

Definition 2.48 (An algebra for partial functions)

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 \(X \setminus \dom f\).

Definition 2.49 (Partial order on real valued (total) functions)

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:

\[ f \preceq g \iff \dom f = \dom g \text{ and } f(x) \leq g(x) \Forall x \in \dom f. \]

Definition 2.50 (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.

Proposition 2.21

A real valued function is bounded if and only if it is bounded from above as well as below.

See also

  1. The set of bounded (total) functions can be turned into a metric space. See Example 3.20.

2.5.2. 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.

2.5.3. Epigraph#

Definition 2.51 (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.

Definition 2.52 (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.

Theorem 2.32 (Epigraph of pointwise maximum of two functions)

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

\[ \epi h = \epi f \cap \epi g. \]

Proof. We first show that \(\epi h \subseteq \epi f \cap \epi g\).

  1. Let \((x, t) \in \epi h\).

  2. Then \(x \in \dom h\) and \(h(x) \leq t\).

  3. Hence \(x \in \dom f\), \(x \in \dom g\), \(f(x) \leq t\) and \(g(x) \leq t\).

  4. Hence \((x,t) \in \epi f\) and \((x, t) \in \epi g\).

  5. 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 \((x, t) \in \epi f \cap \epi g\).

  2. Then \((x, t) \in \epi f\) and \((x, t) \in \epi g\).

  3. Thus \(x \in \dom f\), \(f(x) \leq t\), \(x \in \dom g\) and \(g(x) \leq t\).

  4. Thus \(x \in \dom f \cap \dom g = \dom h\).

  5. Also, \(h(x) = \max(f(x), g(x)) \leq t\).

  6. Hence \((x, t) \in \epi h\).

This result can be generalized for an arbitrary family of functions.

Theorem 2.33 (Epigraph of pointwise maximum of a family of functions)

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

\[ \epi h = \bigcap_{i \in I} \epi f_i. \]

Proof. We first show that \(\epi h \subseteq \bigcap_{i \in I} \epi f_i\).

  1. Let \((x, t) \in \epi h\).

  2. Then \(x \in \dom h\) and \(h(x) \leq t\).

  3. Hence \(x \in \dom f_i\) and \(f_i(x) \leq t\) for every \(i \in I\).

  4. Hence \((x,t) \in \epi f_i\) for every \(i \in I\).

  5. Hence \((x, t) \in \bigcap_{i \in I} \epi f_i\).

The argument for the converse is similar and left as an exercise.

2.5.4. Sub-level Sets#

Definition 2.53 (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

\[ \sublevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \leq \alpha \}. \]

2.5.5. Contours or Level Sets#

Definition 2.54 (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

\[ \contour(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) = \alpha \}. \]

2.5.6. Hypograph#

Definition 2.55 (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.

2.5.7. Super-level Sets#

Definition 2.56 (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

\[ \superlevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \geq \alpha \}. \]

2.5.8. Extended Real Valued Functions#

Definition 2.57 (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 \(\ERL = [-\infty, \infty]\).

Definition 2.58 (Effective domain of an extended real valued function)

For an extended valued function \(\tilde{f} : X \to \ERL\), its effective domain is defined as:

\[ \dom \tilde{f} \triangleq \{ x \in X \ST \tilde{f}(x) < \infty \}. \]

Definition 2.59 (Graphs and level sets)

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.

\[\begin{split} & \graph f \triangleq \{ (x,t) \in X \times \ERL \ST x \in \dom f, f(x) = t \};\\ & \epi f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, f(x) \leq t \};\\ & \epi_s f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, f(x) < t \};\\ & \sublevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \leq \alpha \}; \\ & \contour(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) = \alpha \};\\ & \hypo f \triangleq \{ (x,t) \in X \times \RR \ST x \in \dom f, t \leq f(x) \};\\ & \superlevel(f, \alpha) \triangleq \{ x \in \dom f \,|\, f(x) \geq \alpha \}. \end{split}\]

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 \(\RR\).

  2. If \(f(x) = -\infty\) then \((x, -\infty) \in \graph f\). However \((x, -\infty) \notin \epi f\). But \((x, t) \in \epi f\) for every \(\RR\).

Definition 2.60 (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

\[\begin{split} \tilde{f}(x) \triangleq \begin{cases} f(x) & \text{for} & x \in \dom f \\ \infty & \text{for} & x \notin \dom f \end{cases} \end{split}\]

The extension is pretty useful in analysis and optimization as it extends the domain to the whole of \(X\).

Definition 2.61 (Indicator functions)

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:

\[\begin{split} \tilde{I}_C(x) \triangleq \begin{cases} 0 & x \in C \\ \infty & x \notin C. \end{cases} \end{split}\]