Continuity Over an Interval
Now that we have explored the concept of continuity at a point, let’s extend it to continuity over an interval. A function is continuous over an interval if you can trace it without lifting your pencil between any two points within that interval.
- Open Interval: A function is continuous on an open interval [latex](a,b)[/latex] if it is continuous at every point within that interval.
- Closed Interval: A function is continuous on a closed interval [latex][a,b][/latex] if it is continuous on [latex](a,b)[/latex], continuous from the right at [latex]a[/latex], and continuous from the left at [latex]b[/latex].
continuity over an interval
- A function is continuous over an open interval if it is continuous at every point in the interval.
- A function [latex]f(x)[/latex] is continuous over a closed interval of the form [latex][a,b][/latex] if it is continuous at every point in [latex](a,b)[/latex] and is continuous from the right at [latex]a[/latex] and is continuous from the left at [latex]b[/latex].
- Analogously, a function [latex]f(x)[/latex] is continuous over an interval of the form [latex](a,b][/latex] if it is continuous over [latex](a,b)[/latex] and is continuous from the left at [latex]b[/latex].
Requiring that [latex]\underset{x\to a^+}{\lim}f(x)=f(a)[/latex] and [latex]\underset{x\to b^-}{\lim}f(x)=f(b)[/latex] ensures that we can trace the graph of the function from the point [latex](a,f(a))[/latex] to the point [latex](b,f(b))[/latex] without lifting the pencil. If, for example, [latex]\underset{x\to a^+}{\lim}f(x)\ne f(a)[/latex], we would need to lift our pencil to jump from [latex]f(a)[/latex] to the graph of the rest of the function over [latex](a,b][/latex].
How To: Determine Continuity Over an Interval
- Check Continuity on Open Interval: Verify the function is continuous at all points within [latex](a, b)[/latex].
- Check Right Continuity at [latex]a[/latex]: Ensure [latex]\lim_{x \to a^+} f(x) = f(a).[/latex]
- Check Left Continuity at [latex]b[/latex]: Ensure [latex]\lim_{x \to b^-} f(x) = f(b)[/latex].
State the interval(s) over which the function [latex]f(x)=\dfrac{x-1}{x^2+2x}[/latex] is continuous.
State the interval(s) over which the function [latex]f(x)=\sqrt{4-x^2}[/latex] is continuous.