Real Analysis 33 | Some Continuous Functions [dark version]

TL;DR

Euler’s number e equals exp(1) and can also be expressed as the limit (1+1/n)^n.

Briefing Cornell Notes

Briefing

Exponential and logarithm functions sit at the center of real analysis because they turn addition into multiplication—and that single structural trick drives many of their key properties. The exponential function, written as exp(x), is defined for every real x by a power series. Evaluating it at x = 1 gives Euler’s number e, which can also be characterized by the limit (1 + 1/n)^n. A fundamental identity links exp to itself: exp(x + y) = exp(x)·exp(y). This identity explains the “exponential” naming and leads to the general rule exp(x) = e^x.

From the same foundational setup, exp(x) inherits three properties that matter for later analysis: it is continuous for all real numbers, it is strictly increasing (larger inputs always produce larger outputs), and its end behavior is unbounded above while staying positive—exp(x) → ∞ as x → ∞ and exp(x) → 0 as x → −∞. Because exp maps the interval (0, ∞) onto the positive reals in a one-to-one way, it has an inverse. That inverse is the logarithm function log(x), which is also strictly increasing and continuous. The most important relationship for log comes by flipping the exponential identity: log(ab) = log(a) + log(b). In other words, multiplication inside becomes addition outside, a rule that underpins many transformations in calculus and beyond.

The discussion then broadens from familiar polynomials to a wider class of functions built from infinite sums: power series. Polynomials are continuous because they are finite combinations of powers of x, but power series replace “finitely many coefficients” with “infinitely many coefficients,” producing functions whose domain may be smaller than all real numbers. A power series is defined by choosing a sequence of coefficients (a_k) and summing a_k x^k; the function exists exactly where that series converges. For any such series, there is a maximal radius R (possibly infinite) such that the series converges on the open interval (−R, R). On that interval, the resulting power series defines a continuous function.

A famous example is the sine function. By selecting coefficients that alternate between 0 and ±1/(odd factorials)—with the pattern 0, 1/1!, 0, −1/3!, 0, 1/5!, …—the resulting power series converges for every real x, giving sin(x) with domain all of R. More generally, the radius R can be computed using the root criterion: R is determined by the limit superior of the K-th root of |a_k|, yielding 1/R = limsup_{k→∞} (|a_k|)^{1/k}. This result is known as the Cauchy–Hadamard theorem. With these continuity and convergence foundations in place, the material sets up the next step: differentiating functions defined through such series.

Cornell Notes

Exponential functions exp(x) are defined for all real x using a power series and satisfy exp(x+y)=exp(x)exp(y). This identity yields exp(x)=e^x, where e is the value at x=1 and can also be written as the limit (1+1/n)^n. exp(x) is continuous everywhere, strictly increasing, and tends to ∞ as x→∞ while approaching 0 as x→−∞. Because exp is a bijection from R to (0,∞), its inverse is the logarithm log(x), which is also continuous and strictly increasing, with the key rule log(ab)=log(a)+log(b). The discussion then generalizes from polynomials to power series, which converge on (−R,R) and define continuous functions there; the Cauchy–Hadamard theorem computes R via a limsup root test.

Why does exp(x+y)=exp(x)exp(y) matter beyond being a neat algebraic identity?

It is the structural reason exp turns addition in the exponent into multiplication of values. Setting y=1 repeatedly gives exp(x)=exp(1)^x, so exp(x)=e^x. Taking inverses converts the same structure into log(ab)=log(a)+log(b), turning multiplication into addition—an identity that drives many transformations in calculus.

What end behavior distinguishes exp(x) as x→∞ and x→−∞?

exp(x) grows without bound as x→∞, so exp(x)→∞. Yet exp(x) stays positive for all x and shrinks toward 0 as x→−∞, so exp(x)→0 in that limit.

How does the logarithm inherit monotonicity and continuity from the exponential function?

Since log is the inverse of exp, it reverses the mapping while preserving order: because exp is strictly increasing, log is also strictly increasing. Continuity follows from the inverse-function relationship on the interval where exp is bijective, so log is continuous on (0,∞).

What determines where a power series defines a function?

A power series ∑ a_k x^k defines a function exactly at those real x where the series converges. For any chosen coefficient sequence (a_k), there is a maximal radius R such that the series converges on the open interval (−R,R). Outside that interval, convergence may fail.

How is the sine function obtained from a power series?

Choose coefficients a_0=0, a_1=1/1!, a_2=0, a_3=−1/3!, a_4=0, a_5=1/5!, and continue with zeros on even indices and alternating signs on odd indices using odd factorials. The resulting power series converges for every real x, producing sin(x) with domain all of R.

How does the Cauchy–Hadamard theorem compute the radius of convergence R?

It uses the root test: 1/R equals limsup_{k→∞} (|a_k|)^{1/k}. If that limsup is 0, R is infinite; if it is large, R becomes small. The theorem identifies the largest R for which the interval (−R,R) lies entirely within the domain of convergence.

Review Questions

What identities connect exp and log, and how do they translate addition vs multiplication?
Given a power series with coefficients a_k, how do you compute its radius of convergence using limsup?
Why does a power series define a continuous function on (−R,R), even though it may not converge outside that interval?

Key Points

1
Euler’s number e equals exp(1) and can also be expressed as the limit (1+1/n)^n.
2
The exponential function satisfies exp(x+y)=exp(x)exp(y), which implies exp(x)=e^x.
3
exp(x) is continuous for all real x, strictly increasing, and approaches ∞ as x→∞ while approaching 0 as x→−∞.
4
The logarithm log(x) is the inverse of exp on (0,∞), so it is strictly increasing and continuous, with log(ab)=log(a)+log(b).
5
Polynomials are continuous because they are finite sums of powers, while power series may have restricted domains determined by convergence.
6
Every power series has a maximal radius of convergence R such that it converges (and defines a continuous function) on (−R,R).
7
The Cauchy–Hadamard theorem gives 1/R = limsup_{k→∞} (|a_k|)^{1/k} for computing the radius of convergence.

Highlights

exp(x+y)=exp(x)exp(y) is the engine behind both exp(x)=e^x and log(ab)=log(a)+log(b).

exp(x) stays positive for all x, yet it tends to 0 as x→−∞ and grows without bound as x→∞.

Power series define functions only where the infinite sum converges; that convergence region is controlled by a radius R.

The sine function emerges from a power series with coefficients 0 on even powers and alternating ±1/(odd factorials) on odd powers.

The Cauchy–Hadamard theorem computes the radius of convergence using a limsup root formula.

Topics

Exponential Function
Logarithm Function
Power Series
Radius of Convergence
Cauchy–Hadamard Theorem