To investigate whether some functions are injective and/or surjective, we can typically make our lives easier than working with the raw definitions. In this exercise, we turn to two helpful results which we will use in the exercises to follow.
(i) Show that if a function where is strictly monotonous, i.e. or , then is injective, i.e. that .
A further fact that is helpful also outside the invertability context is the following:
Theorem: Intermediate Value Theorem.
Let for some set , and assume that is continuous. Then, for any with and (and ), for any (for any ), there exists with .
Verbally, this theorem relates to the intuition of being able to draw continuous functions without lifting the pen: if the continuous function attains two different values within the codomain, it will also reach every value in between along the way. The exercise to follow extends this intuition by establishing that for two continuous functions, if one lies above the other at one point but below at another point, then the functions must intersect in between the points.
(ii) Use the intermediate value theorem to show that if two continuous functions with domain and codomain are such that and for some , then there exists a value in between and (i.e., when and else) such that .
(iii) Is inversion of a function a linear operation? That is, does it hold for any two arbitrary, invertible functions with the same domain and codomain that ? Give a formal argument if you think this is true, or a counterexample otherwise. Also think about whether is always guaranteed to be invertible when and are bijective functions.
(i) Consider the function . Is this function invertible?
Hint. The results derived in Ex. 1.a.i and 1.a.ii (or alternatively, the intermediate value theorem) may be helpful in investigating injectivity and surjectivity.
(ii) Consider the function . Is this function invertible? If so, can you determine the inverse?
(iii) Consider the function . Is this function invertible? If so, can you determine the inverse?
For univariate real-valued functions, if they are differentiable, it is commonplace to view the sign of the derivative as an equivalent condition for monotonicity. While this is justified for the non-strict versions, it is indeed not the case that for a differentiable function , , there is equivalence between () and being a strictly increasing (decreasing) function. The reason for this are so-called “saddle points” (as will be thoroughly discussed in Chapter 4) which can occur for strictly monotonous functions and feature .
Now for the task of this exercise: show that for a differentiable function , , is a sufficient, but not a necessary condition for being strictly monotonically increasing. For the latter point, you may consider the function with mapping rule as a counterexample to necessity.
Hint 1: Recall the formal definition of strict monotonicity: , , is strictly monotonically increasing if .
Hint 2: To compare values of using the derivative, recall that for , , and that for a function with for an interval of non-zero length and an “exception set” that contains at most finitely many values, it holds that .
Remark: Showing that for a differentiable function , , is a sufficient, but not a necessary condition for being strictly monotonically decreasing can be done in perfect analogy to the investigation here. To avoid tedious case distinctions, we just focus on the case of strictly monotonically increasing functions in this exercise.
Are the following sets convex? Justify your answer!
(i) Investigate the following function with respect to (strict) convexity/concavity:
Hint: Recall that we can use the second derivative to investigate convexity.
(ii) Investigate the following function with respect to (strict) convexity/concavity:
where is a norm on , .
Hint 1: We know that norms are continuous, but they need not be differentiable. Hence, the criterion for the second derivative is not useful here, and it is instructive to proceed with the “raw” definition of convexity.
Hint 2: If you already solved Exercise 2.a.3., this solution may be helpful here.
(iii) Is the following function convex? Is it quasi-convex?
(i) Consider a univariate monotonically increasing, convex “transformation function” and a convex, potentially multivariate function , . Answer the following:
Summarize your conclusions.
(ii) Can you use (i) to say something about convexity of the function ? Is this consistent with the Hessian criterion?
Hint: Think about the Euclidean norm.
Here, we consider one example of a first and second order multivariate derivative. More exercises can be found on Problem Set 3 and in the examples given in Chapter 3.
Consider the function
You can take for granted that as a composition of infinitely many times differentiable functions (polynomial, logarithm and exponential function), is infinitely many times differentiable. Compute the first and second derivative of , and evaluate them at .
Hint: For the Hessian, you can reduce the number of computations by exploiting symmetry and certain interrelationships of the first order partial derivatives.
Consider a matrix , .
(i) Show that .
(ii) What is the derivative of ?
Hint: Use (i) and the multivariate product rule.
(iii) If , can you find values for and so that the second derivative of is positive definite everywhere? Can you find an alternative combination where is positive semi-definite but not positive definite?
In this exercise, we consider univariate Taylor Approximations. Exercises for the multivariate case can be found on Problem Set 3. Using the univariate case, we can reduce intensity of notation a bit, and focus on getting familiar with the mechanics, investigate approximation quality and study some further properties.
(i) Compute the first and second order Taylor Approximations to the exponential function around and . For , illustrate the exponential function and its approximations. Is one globally preferable to the other, i.e. does it yield a weakly superior approximation everywhere?
(ii) Compute the -th order Taylor Approximation to the exponential function for for variable . Can you find an infinite sum representation for the exponential function using polynomial terms?
Integrate the following functions over the given interval: