Continuity (Differential Calculus)

These are my complete notes for Continuity in Differential Calculus.

I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.

# Continuous Functions: A function is continuous on an interval if it is continuous at every of the interval. A continuous function is a function that is continuous everywhere on its domain.

# Rule 122. Any function y = f(x) whose graph y = f(x) can be sketched in one continuous motion without lifting the pencil is an example of a continuous function.

# Rule 123. There are two RULES of CONTINUITY, which if both met, establish UNIVERSAL CONTINUITY for a function.

A function y = f(x) is continuous at an inerior point c of its domain if:
$$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$$ $$\lim_{x \to c} f(x) = f(c)$$

A function y = f(x) is continuous at a left endpoint a if:
$$\lim_{x \to a^+} f(x) = f(a)$$ and is continous at a right endpoint b of its domain if: $$\lim_{x \to b^-} f(x) = f(b)$$ If a function f is not continuous at a point x, we say that f is discontinuous at c, and that c is a "point of discontinuity" of f.

# Rule 124. Continuous Function are continuous everywhere on their domains. Properties of Continuous Functions f & g are continuous at x = c.

All polynomial and Radical functions are continuous everywhere with no Points of Discontinuities. On the other hand, all rational and trig. functions are only continuous along their domain; they have Points of Discontinuities wherever the denominator equals 0.

# Operation Terminology Refresher:

Sums: f + g

Differences: f - g

Products: f × g

Constant Multiples: k × f, where k = any real #.

Quotients: f/g, g ≠ 0.

# Rule 125. If f is continuous at c, and g is continuous at f(c), then the composite g ∘ f is continuous at c.

# Rule 126. A function is continuous even if it has Points of Discontinuity, as long as these points are not within the domain. For example, 1/x is considered to be continuous along its domain, despite the absurdity of its graph.

# Rule 127. A function passes the Intermediate value theorem if it never takes on two values without taking on every value in between. A continuous function f on a closed interval [a, b] will take on every y-value between f(a) and f(b), given no discontinuities.

# Rule 128. Jump Discontinuities: Where the left & right side limits are not equal.

Infinite Discontinuities: Where the discontinuity is an asymptote.

Removable Discontinuities: A detached dot, floating on the plane, where the limit continues to be a real number.

Oscillating Discontinuities: A discontinuity that oscillates back and forth, son. Examples include sin(1/x). These functions are discontinuities because no matter how far you zoom in, it will still be oscilliating, thus lacking local linearity.

# Rule 129. For Discontinuities: Jump beats hole. Infinite beats Jump. Oscillating is in a league of its own, as it can only be sin(1/x) or something of the sort.

Summary of Continuity (Differential Calculus)

?. Continuity.