Almost every function anyone writes down is continuous, and the reason is that continuity survives every way of building one function out of others. Sums, products and quotients inherit it from the algebra of limits, and composition inherits it by a chain of two radii: the outer function names a tolerance, the inner one answers with a radius, and that radius becomes the tolerance the outermost step has to meet.

Two machines in series. To hold the final output within a tolerance, ask the second machine what accuracy it needs on its input, and then ask the first machine to deliver that. The answer to the second question is a tolerance for the first, and the chain can be made as long as you like.

If $f$ and $g$ are continuous at $c$ then so are $f\pm g$, $fg$ and $kf$, and so is $\tfrac fg$ provided $g(c)\ne0$. These follow from the algebra of limits. Composition is genuinely new: if $f$ is continuous at $c$ and $g$ is continuous at $f(c)$, then $g\circ f$ is continuous at $c$. Note carefully where the second function must be continuous — at the image of the point, not at the point.

• Every polynomial is continuous everywhere, and every rational function is continuous wherever its denominator is not zero.
• $\lvert f\rvert$, $\max(f,g)$ and $\min(f,g)$ are continuous whenever $f$ and are, since .

Work from the outside in. A tolerance on the final output is handed to the outer function, which is continuous at the image of the point and so names a radius that keeps its own output inside that tolerance. That radius is a tolerance on the inner output, so hand it to the inner function, which is continuous at the point and names a radius of its own. Following the two steps forward shows that radius does what was asked.

A tolerance on the final output.

The outer function is continuous at the image of the point, so it names a radius.

That radius is a tolerance for the inner function, which is continuous at the point.

Follow the two implications in order.

Which is the definition, for the tolerance given.

The outer function has to be continuous where the inner one sends the point.

The composition rule needs continuity of the outer function. With limits alone it fails.

• Sums, products, constants and quotients with a non-zero denominator inherit continuity.
• A composition is continuous when the outer function is continuous at the image of the point.
• The same statement for limits alone is false.
• $\lvert f\rvert$, $\max$ and $\min$ are continuous, because they are formulas in continuous functions.

Great work! You now know how to:

• Apply the combining rules to build continuous functions
• Say where the outer function of a composition must be continuous
• Give the counterexample showing that limits alone do not compose
• Recognise that almost every written formula is continuous, and say why