Real Analysis 29 | Combination of Continuous Functions [dark version]

TL;DR

Continuity is a local property: checking behavior near x0 is enough to determine continuity at x0.

Briefing Cornell Notes

Briefing

Continuity behaves well under standard algebraic operations and under function composition: combine continuous functions and the result stays continuous (with a small extra condition for division). The core idea is local—continuity at a point x0 means small changes in the input near x0 produce only small changes in the output near f(x0)—and the transcript uses that local viewpoint to build a set of “closure” rules that avoid re-checking the full definition every time.

First come the operations. If two functions f and g share the same domain and are continuous at the same point x0, then their sum f+g is continuous at x0 and their product f·g is continuous at x0. The justification can be done cleanly using the sequence-based definition of continuity: once xn→x0, the limit theorems for sequences let the limit pass through addition and multiplication. Division works similarly, but only after excluding the problematic case: if g(x0)≠0, then the quotient f/g is continuous at x0. The domain may need adjustment to ensure division is defined, but the key extra requirement is that the denominator does not vanish at the point where continuity is being tested.

Next is composition, where one function feeds into another. With functions G and F, the composition is written as F∘G (read right-to-left: apply G first, then F). Continuity survives composition provided the functions “fit” together: G must map the domain J into the domain I of F, meaning the image of J under G lies inside I. Under that compatibility condition, if both G and F are continuous at the relevant points—G at x0 in J, and F at G(x0) in I—then F∘G is continuous at x0. The transcript also distinguishes the point-level input for continuity (x0) from the set-level domain restriction (J and I), emphasizing that composition only makes sense when G(x) stays within the region where F is defined.

A sequence proof is sketched for composition: take any sequence xn in J with xn→x0. Continuity of G gives G(xn)→G(x0). Then continuity of F at G(x0) yields F(G(xn))→F(G(x0)). Since F(G(x0)) is exactly (F∘G)(x0), the limit criterion matches the definition of continuity at x0. Finally, the results extend from a single point to all points: if f and g are continuous everywhere on their domains (and the domain conditions for composition hold), then the sum, product, quotient (where allowed), and composition remain continuous everywhere.

These closure properties matter because continuity is a gateway to many stronger “regularity” behaviors studied later. Instead of re-deriving continuity from scratch repeatedly, the algebra and composition rules let continuous building blocks be assembled into more complex functions without losing continuity.

Cornell Notes

Continuity at a point x0 is preserved under addition and multiplication: if f and g are continuous at x0 (and share a domain), then f+g and f·g are continuous at x0. Continuity also survives division as long as g(x0)≠0 and the quotient is defined near x0. For composition, continuity is preserved when the functions are compatible: G must map the domain J into the domain I of F, and continuity of G at x0 together with continuity of F at G(x0) implies F∘G is continuous at x0. Sequence-based proofs make these results straightforward by passing limits through algebraic operations and through nested functions.

Why does continuity of f and g imply continuity of f+g and f·g at the same point x0?

Both operations are built from applying algebra to the output values. Using the sequence definition: for any sequence xn→x0, continuity gives f(xn)→f(x0) and g(xn)→g(x0). Limit theorems for sequences then yield (f(xn)+g(xn))→f(x0)+g(x0) and (f(xn)g(xn))→f(x0)g(x0). That matches the definition of continuity at x0 for f+g and f·g.

What extra condition is needed to claim continuity of the quotient f/g at x0?

The denominator must not vanish at the point of interest. The transcript’s condition is g(x0)≠0, because division is only valid where g(x) stays away from zero at least at x0 (and the quotient must be defined on the relevant domain). With g(x0)≠0, the same limit-passing logic works: if xn→x0, then f(xn)/g(xn)→f(x0)/g(x0).

When does the composition F∘G make sense, and how does that affect continuity?

Composition requires domain compatibility: G must map the domain J into the domain I of F. In set terms, the image of J under G must be a subset of I, so that F(G(x)) is defined for every x in J. Continuity then depends on the correct “input points”: G must be continuous at x0∈J, and F must be continuous at the point G(x0)∈I.

How does the sequence proof establish continuity of F∘G at x0?

Pick a sequence xn in J with xn→x0. Continuity of G at x0 gives G(xn)→G(x0). Then continuity of F at G(x0) gives F(G(xn))→F(G(x0)). Since F(G(x0)) equals (F∘G)(x0), the limit criterion for continuity at x0 is satisfied.

How do pointwise continuity results extend to continuity on entire domains?

The closure rules are formulated at a single point x0. If f and g are continuous at every point in their domains (and the quotient/composition domain conditions hold), then the same reasoning applies at each point. That lifts the results from “continuous at x0” to “continuous everywhere on the domain.”

Review Questions

Suppose f and g are continuous at x0 and share a domain. Which operations preserve continuity automatically, and which operation requires an additional condition?
For composition F∘G, what set-inclusion condition ensures the composition is defined, and which two points must be checked for continuity?
Using the sequence definition, outline the key limit steps needed to prove continuity of F∘G at x0.

Key Points

1
Continuity is a local property: checking behavior near x0 is enough to determine continuity at x0.
2
If f and g are continuous at x0 (with a shared domain), then f+g and f·g are continuous at x0.
3
The quotient f/g is continuous at x0 provided g(x0)≠0 and the quotient is defined near x0.
4
Composition F∘G preserves continuity when G maps the domain J into the domain I of F (so F(G(x)) is always defined).
5
For continuity of F∘G at x0, it’s enough that G is continuous at x0 and F is continuous at G(x0).
6
Sequence-based proofs work by pushing limits through algebra and through the nested structure of composition.

Highlights

Addition and multiplication of continuous functions stay continuous at the same point x0.

Division preserves continuity only when the denominator is nonzero at the point: g(x0)≠0.

For composition, continuity survives as long as G’s outputs stay inside F’s domain and both continuity checks happen at x0 and at G(x0).

Topics

Continuity
Function Operations
Quotient Rule
Composition
Sequence Definition