Discrete vs Continuous

Discrete vs continuous is one of the most consequential distinctions in mathematics. Misidentifying which you have leads to wrong tools, wrong distributions, wrong conclusions.

Discrete

A discrete quantity can take only separated values, usually integers or a finite set.

Examples: number of students in a class, dice roll outcomes, defects per unit, clicks on a webpage.

Math tools: summation $\sum$, probability mass functions $P(X = k)$, combinatorics, difference equations, graph theory.

Continuous

A continuous quantity can take any value within a range, with arbitrary precision.

Examples: height, weight, time, temperature, distance.

Math tools: integration $\int$, probability density functions $f(x)$ (where $P(X = \text{exact value}) = 0$), differential equations, calculus.

The decision: which framework?

Aspect	Discrete	Continuous
Values	Separate, countable	Range, uncountable
Probability of exact value	$P(X = k) > 0$	$P(X = a) = 0$ — must use intervals
"Sum" tool	$\sum$	$\int$
Equation type	Difference equation	Differential equation
Common distributions	Binomial, Poisson, geometric	Normal, exponential, uniform

Common mistakes

• Treating counts as continuous. "Average household has 2.3 children" — fine for summary, but probability of "exactly 2.3 children" is meaningless.
• Treating measurements as discrete. Height "is 170 cm" rounds a continuous quantity; statistical tests assuming discreteness lose information.
• Mixing in probability: don't sum a continuous PDF; integrate. Don't integrate a discrete PMF; sum.

Bridges between

The central limit theorem lets discrete sums of many small variables approximate a continuous normal. The continuity correction translates between binomial (discrete) and normal (continuous) probabilities. Riemann sums are the discrete bridge to integrals.