Can you draw the graph without lifting your pen? Intuitively, a function is continuous when its graph flows without breaks, holes, or sudden jumps. A real-world offer makes the jump easy to see — and easy to read carefully.

Example A discount that jumps

A supermarket offers a discount based on how much you spend: less than 300 reals gets no discount; 300 up to 600 reals gets a 40-real discount; more than 600 reals gets a 100-real discount. Graphed, the discount sits flat at 0, jumps to 40 at 300, holds there, then jumps again to 100 at 600.

At exactly 300, the store's wording decides: does 300 belong with the lower group or the upper one?

At exactly 600, the same question repeats — and it is the one the lesson focuses on.

Concept Reading open and filled points

A filled point marks the value the function actually takes there. An open (hollow) point marks a value the graph approaches but does not attain — that input belongs to a different piece of the rule.

If the 100-real discount applies only to purchases above 600, a purchase of exactly 600 still gets 40: filled at 40, open at 100 — the figure above. If instead 600 or more gets the larger discount, the two points swap: open at 40, filled at 100.

Either way, someone spending exactly 600 reals gets one specific discount — never both, and never something in between.

Note Still a function, just not continuous

Whichever policy the store chooses, every purchase amount still produces exactly one discount — so this is a perfectly good function. It simply is not a continuous one: its graph contains sudden jumps, and no single pen stroke can trace it.

Concept Applying the three-part test at 600

A function is continuous at a point when it is defined there, its limit exists there, and the two agree. Check the discount function at 600, under the first policy (filled at 40):

Defined at 600? Yes — the discount is 40.

Approaching from the left: the discount heads toward 40.

Approaching from the right: the discount heads toward 100.

⟹ the two one-sided limits disagree, so no limit exists at 600

The function fails the second condition, so it is discontinuous at 600 — regardless of which value happens to be filled in. Filling in a point only decides which value the function takes there, never whether the jump itself disappears.

Summary

A function is continuous where its graph can be traced without lifting the pen.
A filled point is the value the function takes; an open point is a value only approached, never attained.
The store's exact wording decides which point is filled — but either choice still defines a function.
A jump discontinuity happens where the left-hand and right-hand limits disagree.
Filling in one point can never repair a jump — only a removable hole can be fixed that way.

Continuity and the Supermarket Discount