Continuity
Continuity
Objectives
- Explain the three conditions for continuity at a point.
- Describe three kinds of discontinuities.
- Define continuity on an interval.
- State the theorem for limits of composite functions.
- Provide an example of the intermediate value theorem.
Summary
A function \(f\) has limit \(L\) as \(x \to a\) if and only if \(f\) has a left-hand limit at \(x = a\text{,}\) \(f\) has a right-hand limit at \(x = a\text{,}\) and the left- and right-hand limits are equal. Visually, this means that there can be a hole in the graph at \(x = a\text{,}\) but the function must approach the same single value from either side of \(x = a\text{.}\)
A function \(f\) is continuous at \(x = a\) whenever \(f(a)\) is defined, \(f\) has a limit as \(x \to a\text{,}\) and the value of the limit and the value of the function agree. This guarantees that there is not a hole or jump in the graph of \(f\) at \(x = a\text{.}\) So we say that 𝑓 is continuous at point 𝑥=𝑎 whenever
lim𝑥→𝑎 𝑓(𝑥) = 𝑓(𝑎)
that is to say that the following conditions are met:
1. lim𝑥→𝑎− 𝑓(𝑥) = 𝑓(𝑎) and
2. lim𝑥→𝑎+ 𝑓(𝑥) = 𝑓(𝑎).
A function \(f\) is differentiable at \(x = a\) whenever \(f'(a)\) exists, which means that \(f\) has a tangent line at \((a,f(a))\) and thus \(f\) is locally linear at \(x = a\text{.}\) Informally, this means that the function looks like a line when viewed up close at \((a,f(a))\) and that there is not a corner point or cusp at \((a,f(a))\text{.}\)
Of the three conditions discussed in this section (having a limit at \(x = a\text{,}\) being continuous at \(x = a\text{,}\) and being differentiable at \(x = a\)), the strongest condition is being differentiable, and the next strongest is being continuous. In particular, if \(f\) is differentiable at \(x = a\text{,}\) then \(f\) is also continuous at \(x = a\text{,}\) and if \(f\) is continuous at \(x = a\text{,}\) then \(f\) has a limit at \(x = a\text{.}\)
See the Desmos demonstration about Continuity.
See the Desmos demonstration about Continuity of Piecewise Functions.