Free lesson · Differential calculus

Functions, limits and continuity

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Start with the idea

A function turns an input into an output.
A limit asks what the output approaches as the input gets near a point; the function does not have to be defined at that point.
Continuity adds the requirement that the limiting value agrees with the actual value.

Symbols, units & horizon

x: input value
f(x): output at x
lim: limiting value
x→2: approach 2 without requiring x=2
≠: not equal
g: new function agreeing with f away from 2, with g(2)=4 at the missing point
Example x values are dimensionless

When and why to use this

Use limits to understand boundary behaviour and continuity to decide whether small input changes can produce abrupt output changes. These concepts are useful in physics, geometry, engineering and numerical model design.

Start with a concrete mapping: a square of side x has area f(x)=x².
Input units are length and output units are length squared.
The domain specifies allowed inputs.
A formula involving division excludes zero denominators; a real logarithm requires a positive argument.
To investigate a limit, simplify first or approach from both sides.
A table can suggest an answer, but a few sampled values do not prove a limit exists.
A function with a jump has different one-sided limits; a removable hole can have the same limiting value on both sides.

f (x) = \frac{x^{2} - 4}{x - 2} = x + 2 (x \neq 2), \lim_{x \to 2} f (x) = 4

Calculus: derivation and arithmetic

Functions, limits and continuity

Factor the numerator: x²−4=(x−2)(x+2). For x≠2, cancel the nonzero factor x−2.
The remaining expression x+2 approaches 4 as x approaches 2 from either side. The original quotient is still undefined at x=2.
Define a new function g with the same values away from 2 and g(2)=4. That extension is continuous at 2.

Work it by hand

At x=1.9 the quotient is 3.9; at x=2.1 it is 4.1. The limit is 4, while substituting 2 into the unsimplified quotient gives the undefined expression 0/0.

An analogy to remember

A road can approach the same height from both directions even if one paving stone is missing. The limit describes the approach; continuity asks whether the stone is present at the matching height.

How this becomes a building block

Limits are the foundation of derivatives and definite integrals. In a trading system, checking a model’s domain can prevent invalid calculations at zero time, zero volatility or zero liquidity. A removable singularity may need a carefully derived limiting implementation rather than a divide-by-zero workaround.

Further reading: OpenStax Calculus, Volume 1 — definitions and standard theorems ↗

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def quotient_with_hole(x):
    if x == 2:
        raise ValueError("Original function undefined at x=2")
    return x + 2  # algebraically equivalent on the original domain

def continuous_extension(x):
    return x + 2

print(quotient_with_hole(1.9), continuous_extension(2))

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