7.2 Continuity

A function is continuous where its graph can be drawn through without lifting the pen. Put precisely, it comes down to three conditions.

Definition 7.12. A function \(f\) is continuous at \(x=a\) if all three of the following hold:

(i).: \(f(a)\) is defined;
(ii).: \(\lim _{x\to a}f(x)\) exists;
(iii).: \(\lim _{x\to a}f(x)=f(a)\).

If any one of them fails, \(f\) is discontinuous at \(x=a\).

Note 7.13. The three conditions are in order and each depends on the ones before it. There is no point comparing a limit to a value that does not exist, so a function failing (i) fails outright, and (iii) can only be asked once (i) and (ii) have both been passed.

Example 7.14. Discuss the continuity of each function at the value given.

(a) \(g(x)=\begin {cases} x^2+3 & \text {if } x\neq 1,\\ 10 & \text {if } x=1,\end {cases}\) at \(x=1\).

(b) \(f(x)=\begin {cases} x^2-1 & \text {if } x\leq 3,\\ 2x+1 & \text {if } x>3,\end {cases}\) at \(x=3\).

Solution. (a) Taking the conditions in turn: \[\text {(i) } g(1)=10,\ \text {defined};\qquad \text {(ii) } \lim _{x\to 1}g(x)=4,\ \text {exists};\qquad \text {(iii) } 4\neq 10.\] The first two hold and the third fails, so \(g\) is discontinuous at \(x=1\). The break is a single misplaced point, and moving it from \(10\) down to \(4\) would repair the function completely.

(b) \(f(3)=9-1=8\), so (i) holds. But from the earlier example the one-sided limits are \(8\) and \(7\), so the limit does not exist and (ii) fails. Hence \(f\) is discontinuous at \(x=3\). Here the graph jumps, and no single repair will close the gap.

(c) At \(x=5\) the denominator is zero, so \(h(5)\) does not exist and (i) fails immediately. \(h\) is discontinuous at \(x=5\).

Note that the limit is perfectly well behaved: \[\lim _{x\to 5}\frac {x^2-25}{x-5}=\lim _{x\to 5}\frac {(x-5)(x+5)}{x-5} =\lim _{x\to 5}(x+5)=10.\] The function simply has a hole at \(x=5\), with nothing in it.

Note 7.15. The three parts fail three different conditions, in reverse order — (a) fails (iii), (b) fails (ii), (c) fails (i) — and that is the whole content of the definition. When answering, name which condition fails rather than just declaring the function discontinuous.

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