Chapter 4 — Key Takeaways

A structured recap of continuity: the definition, the discontinuity zoo, the two great existence theorems, and the algorithm IVT becomes.

1. The Definition of Continuity (Section 4.2)

A function $f$ is continuous at a point $a$ when three conditions all hold:

1. $f(a)$ is defined;
2. $\lim_{x \to a} f(x)$ exists;
3. $\lim_{x \to a} f(x) = f(a)$.

Compactly: continuity is the single equation $\lim_{x\to a} f(x) = f(a)$ — the limit equals the value, "where it heads is where it is." The three conditions are independent; each can fail while the other two hold.

• Continuity is a pointwise property — always say where. "$f$ is discontinuous" is incomplete; "$f$ is discontinuous at $x=2$" is correct.
• Continuous on an interval means continuous at every point of it; continuous (no qualifier) means continuous at every point of the domain.
• The formal $\varepsilon$–$\delta$ form: $\forall\varepsilon>0\ \exists\delta>0$ such that $|x-a|<\delta \implies |f(x)-f(a)|<\varepsilon$, with the target value forced to be $f(a)$ itself.
• The substitution shortcut: for a continuous function, $\lim_{x\to a} f(x) = f(a)$ — you evaluate the limit by plugging in. This is why the hard limit work of Chapter 3 was only needed where functions fail to be continuous.

2. The Classical Continuous Functions (Section 4.2)

Continuous on their natural domains: polynomials (all of $\mathbb{R}$), rational functions (where the denominator $\ne 0$), exponentials $e^x, b^x$ (all of $\mathbb{R}$), logarithms $\ln x, \log_b x$ (on $(0,\infty)$), sine and cosine (all of $\mathbb{R}$), tangent (except where $\cos x = 0$), and roots $\sqrt{x}$ (on $[0,\infty)$, one-sided at $0$).

3. The Four Types of Discontinuity (Section 4.3)

Classifying the failure tells you everything about the local behavior. A single algebraic formula guarantees nothing: $1/x$, $\tan x$, and $\sin(1/x)$ are each one clean formula and each discontinuous somewhere.

4. Continuity of Combinations and Compositions (Section 4.5)

If $f$ and $g$ are continuous at $a$, then so are $f+g$, $f-g$, $cf$, $f\cdot g$, and $f/g$ (provided $g(a)\ne 0$). If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$ — because continuity lets the limit pass inside: $\lim_{x\to a} f(g(x)) = f(\lim_{x\to a} g(x))$.

Consequence: anything you build from the classical list using $+,-,\times,\div$, and composition is automatically continuous on its domain — you certify it by structure, not by checking three conditions point by point. Continuity is the rule; the interesting questions are always where and how a function fails.

5. The Intermediate Value Theorem (Section 4.6)

If $f$ is continuous on $[a,b]$ and $N$ lies strictly between $f(a)$ and $f(b)$, then $f(c)=N$ for some $c\in(a,b)$.

• Root-finding corollary: if $f$ is continuous and $f(a), f(b)$ have opposite signs, then $f$ has a root in $(a,b)$.
• IVT proves existence, never uniqueness, and only one direction: a sign change forces a root, but no sign change does not forbid one.
• IVT is not a free consequence of "no jumps." It fails over $\mathbb{Q}$ (e.g. $x^2-2$ has no rational root despite a sign change); what rescues it over $\mathbb{R}$ is completeness — the real line has no gaps.
• Lose continuity and it dies: a patched $1/x$ on $[-1,1]$ leaps over intermediate values through its infinite discontinuity.

6. The Bisection Method (Section 4.7)

IVT made executable. Given $f$ continuous on $[a,b]$ with a sign change:

1. compute the midpoint $m=(a+b)/2$ and $f(m)$;
2. keep the half across which the sign still flips (replace $a$ or $b$ with $m$);
3. repeat.

• After $n$ steps the bracket has width $(b-a)/2^n$; reaching tolerance $\tau$ takes about $\log_2\!\big((b-a)/\tau\big)$ steps.
• Linear convergence — one bit (~0.3 decimal digit) per step. Slow but guaranteed for any continuous sign-change bracket.
• Robust code tests both $|f(m)|<\tau$ and interval width $<\tau$, and uses $f(a)f(m)<0$ to detect the sign change cleanly.
• Trade-off vs. Newton's method (Chapter 11): Newton converges quadratically (doubles correct digits per step) but needs a derivative and a good guess and can diverge; bisection needs only continuity and a sign change and never fails.

7. One-Sided Continuity and the Extreme Value Theorem (Sections 4.8–4.9)

• $f$ is continuous from the right at $a$ if $\lim_{x\to a^+} f(x)=f(a)$; from the left if $\lim_{x\to a^-} f(x)=f(a)$. An interior point is continuous iff continuous from both sides. "$f$ continuous on $[a,b]$" means continuous at interior points, right-continuous at $a$, left-continuous at $b$.
• Extreme Value Theorem: if $f$ is continuous on a closed, bounded interval $[a,b]$, it attains both an absolute max and an absolute min on $[a,b]$. Both hypotheses are essential — drop continuity ($1/x$ patched at $0$ runs to $+\infty$) or closedness ($f(x)=x$ on $(0,1)$ reaches neither endpoint) and the guarantee collapses.
• EVT, like IVT, secretly rests on completeness (via Bolzano–Weierstrass). It is the foundation that makes optimization (Chapter 10) well-posed: a best value is known to exist before you go hunting for it.

8. Continuity Is the Hypothesis Behind Everything (Section 4.10)

Continuity is the admission ticket to the rest of calculus. It is the quiet hypothesis behind the Mean Value Theorem (Chapter 9), the Fundamental Theorem of Calculus (Chapter 14), Riemann integrability (Chapter 13), and existence/uniqueness for differential equations (Chapter 19). A one-way bridge to prove in Chapter 6: differentiability implies continuity, but not conversely ($|x|$ is continuous yet has a corner at $0$).

Common Errors to Avoid

• Saying a function "is discontinuous" without naming the point. Continuity is pointwise.
• Assuming a single formula must be continuous. Check suspect points: vanishing denominators, piecewise seams, blow-ups.
• Treating IVT as an if-and-only-if: no sign change does not rule out a root.
• Expecting IVT to give a unique root, or to tell you where the root is — it only certifies existence.
• Forgetting that EVT needs both continuity and a closed, bounded interval.
• Confusing continuity of $f$ with continuity of its slope: a continuous function (like a tax schedule, or $|x|$) can have a jumpy derivative.

Skills You Should Now Have

• Test continuity at a point via the three conditions (Section 4.2).
• Classify a discontinuity as removable, jump, infinite, or oscillating (Section 4.3).
• Find parameter values that make a piecewise function continuous at its seams (Sections 4.2, 4.8).
• Certify continuity of a built-up function by structure (Section 4.5).
• Apply IVT to prove a root or intermediate value exists (Section 4.6).
• Run the bisection method by hand and in Python, and predict its iteration count (Section 4.7).
• State EVT and explain why each hypothesis is needed (Section 4.9).

What's Next

Chapter 5 uses limits and continuity to study rates of change — the average rate over an interval sharpening into an instantaneous rate. That limit of secant slopes becomes, in Chapter 6, the derivative. Continuity will stand quietly behind every line of that construction, and the differentiability-implies-continuity bridge promised here gets proved there.