Continuity

The pencil test

If you can draw a graph through a point without lifting your pencil, the graph is continuous there. No holes, no gaps, no sudden jumps. It's a good picture to start with, but "lifting a pencil" isn't something you can check with algebra, so calculus turns it into three precise checks.

The three checks

A function f is continuous at x = a when all three of these are true:

1. f(a) exists. The function has a value at a: there's a filled dot on the graph there.
2. lim x→a f(x) exists. The graph heads to the same height from the left and from the right. (From the last lesson: the two-sided limit exists only when the left-hand and right-hand limits agree.)
3. They're equal. The limit equals f(a): where the graph is heading is exactly where it lands.

If any check fails, the function is discontinuous at a.

Worked example: a piecewise function

Take f(x) = x + 1 for x < 2, and f(x) = 2x − 1 for x ≥ 2. Check a = 2:

1. f(2) uses the piece for x ≥ 2: 2(2) − 1 = 3. It exists. ✓
2. From the left, x + 1 heads to 2 + 1 = 3. From the right, 2x − 1 heads to 2(2) − 1 = 3. Both sides agree, so lim x→2 f(x) = 3. ✓
3. The limit is 3 and f(2) is 3. Equal. ✓

All three checks pass, so f is continuous at x = 2. The two pieces meet at the point (2, 3).

When it jumps

Now change the right piece to 2x for x ≥ 2. At x = 2 it starts at 2(2) = 4, while the left piece still heads to 3.

1. f(2) = 4. It exists. ✓
2. Left-hand limit 3, right-hand limit 4. They don't agree, so lim x→2 f(x) doesn't exist. ✗
3. There's no limit to compare with f(2), so this check can't pass either.

The graph jumps from 3 up to 4 at x = 2. This is a jump discontinuity: you'd have to lift your pencil.

Try it yourself

Let g(x) = 3x for x < 1, and g(x) = x + m for x ≥ 1. What value of m makes g continuous at x = 1?

The left piece heads to 3(1) = 3. The right piece gives g(1) = 1 + m. Set them equal: 1 + m = 3, so m = 2. Check: with m = 2, both sides give 3 at x = 1.

Good to know

• The three checks are the standard definition of continuity at a point x = a (AP Calculus CED; OpenStax Calculus Volume 1, 2.4): f(a) is defined, lim x→a f(x) exists, and the limit equals f(a).
• The "pencil test" is a picture of continuity on an interval, not the definition.
• In the jump version (2x for x ≥ 2), the left-hand limit is 3 and the right-hand limit is 4, so the two-sided limit doesn't exist and check 2 fails, even though f(2) = 4 does exist.
• Removable discontinuities (holes) and infinite discontinuities (vertical asymptotes) are other ways continuity can fail; they aren't covered here, to keep it to one idea.

Answer to the comment question

f is x² when x is under 3, and x + k from 3 on. What k makes it continuous? k = 6. The left piece heads to 3² = 9. The right piece gives f(3) = 3 + k. For the limit to exist and equal f(3), set 3 + k = 9, so k = 6.