APFIVE Applying the Definition of Continuity
Which of the following is an example of a continuous function on a given interval based on the Continuity Conditions?
| x | f(x) |
|---|---|
| -1 | 2 |
| 0 | 1 |
| 1 | 2 |
A
The function $$f(x)=x^2+1$$ is continuous for all $$x$$ since for every point, $$\lim_{x \to c} f(x)=f(c)$$.
B
A piecewise function like $$\begin{cases} x-1 & x<2 \\ x+1 & x\ge2 \end{cases}$$ has a jump discontinuity at $$x=2$$ since the one-sided limits do not match.
C
The function $$f(x)=\frac{1}{x-2}$$ is not continuous at $$x=2$$ because it is undefined there.
D
A function defined by $$f(x)=\begin{cases} \sin(x) & x\leq \pi \\ \cos(x) & x>\pi \end{cases}$$ may fail continuity at the boundary if the values do not agree.
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