Limits and Continuity | Moosa Academy

Differentiation measures a rate of change; integration measures an area. Neither works where a curve tears or jumps. Continuity is the condition that rules that out, and limits are the tool used to test for it — which makes this lesson the doorway into calculus.

Concept The limit

This says: as gets arbitrarily close to , the value of settles on . Note what it does not say — it makes no claim about itself. The function may not even be defined there.

Approaching from the right:

Approaching from the left:

The limit exists only if the two one-sided limits agree. If the curve heads for different values from each side, there is no single value to settle on.

Concept Continuity and its three conditions

A function is continuous if you can draw it without lifting your pen.

Stated precisely, is continuous at when all three hold:

is defined.
exists.
.

Fail any one and the function is discontinuous there.

Note Three kinds of discontinuity

Infinite — the curve runs off to ±∞, as

does at 0.

Removable — a single hole; the limit exists but the value is missing or wrong.

Jump — the one-sided limits differ, so the curve steps.

Only the removable kind can be repaired — redefine the function at that one point and the gap closes.

Example A removable discontinuity

Examine at .

the denominator is zero, so

is undefined — condition 1 already fails.

Factorise:

for every

from both sides, so the limit does exist.

Defining

closes the hole.

⟹ removable discontinuity at x = 2

Example A jump discontinuity

Examine at .

From the left:

From the right:

The one-sided limits differ, so no limit exists at

⟹ jump discontinuity — the curve steps from 2 to 4

No redefinition can fix this: a single value cannot equal both 2 and 4.

Theorem The Intermediate Value Theorem

If is continuous on and lies between and , then for some in the interval.

A continuous curve cannot skip values. To get from one height to another it must pass through everything in between.

and

, the function must cross zero somewhere between — which proves a root exists without finding it.

and

, the function must take the value 3 somewhere between.

Continuity is essential to the argument. A function that jumps can step straight over a value without ever taking it.

Summary

A limit describes where a function heads as x approaches a point, not its value there.
The limit exists only when the left and right one-sided limits agree.
Continuity at c needs f(c) defined, the limit to exist, and the two to be equal.
Discontinuities are infinite, removable or jump — only the removable kind can be repaired.