Continuity
“Continuity is just 'limit value = actual value'.”
The formula
lim(x→a) f(x) = f(a)How to read it: the limit at x=a equals the function's value there — then f is continuous at a
- f(a)
- — the actual value of the function at that point
- lim(x→a) f(x)
- — the limit as you approach that point
- =
- — when the two match, f is continuous at a
The hook
Continuity nails down 'can you draw it without lifting your pencil' into a single equation.
In plain words
If the value you expected on approach (the limit) exactly equals the value actually marked at that point, the curve runs through it unbroken.
The intuition
No holes (missing points), no jumps (steps), no blow-ups (vertical spikes). If you can trace the graph with a fingertip and never lift it, it's continuous.
How it's built
Three things must hold at once — ① f(a) is defined, ② the limit exists, and ③ they're equal. Break any one and the curve tears at that point.
Example
Take f(x)=x²+1 at x=1: the limit lim(x→1)=2 and the value f(1)=2. They agree, so f is continuous at x=1.
Common mistake
Continuous doesn't mean differentiable. |x| is unbroken at x=0 (continuous) but has a sharp corner, so it has no well-defined slope there.
Where it's used
Powerhouse guarantees like the Intermediate Value Theorem and Extreme Value Theorem — 'something must exist somewhere' — only work on continuous functions.
Where it came from
'No breaks' used to be left to intuition; in the 1800s Cauchy defined it via limits, so continuity became something you handle with inequalities, not pictures.
Prerequisites
Quick check
What is required for f to be continuous at x=a?
- the limit equals the function's value✓
- the value must be 0
- the slope must be 0
- f must be differentiable at a
Practice
Is f(x)=2x+1 continuous at x=3? Find the limit and f(3) separately, then compare.
Answer: undefined
- Limit: lim(x→3)(2x+1) = 7
- Value: f(3) = 2·3+1 = 7
- They match, so f is continuous at x=3
Takeaway: Polynomials are continuous everywhere — the limit is just the plugged-in value.
f(x)=x²−x is continuous, so lim(x→2) f(x) = f(2). What is it?
Answer: 2
- Continuous, so just substitute
- f(2) = 4 − 2 = 2
Takeaway: For a continuous function, the limit is a single substitution.
Which of these is NOT continuous?
Answer: 0
- 1/x is undefined at x=0 and blows up to ±∞
- The others run through their points unbroken
Takeaway: Undefined, jump, or blow-up → discontinuous there.