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6.E: Continuity - What It Isn’t and What It Is (Exercises)

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Q1

Use the deﬁnition of continuity to prove that the constant function $$g(x) = c$$ is continuous at any point a.

Q2

1. Use the deﬁnition of continuity to prove that $$\ln x$$ is continuous at $$1$$. [Hint: You may want to use the fact $$\left |\ln x \right | < \varepsilon \Leftrightarrow -\varepsilon < \ln x < \varepsilon$$ to ﬁnd a $$δ$$.]
2. Use part (a) to prove that $$\ln x$$ is continuous at any positive real number $$a$$. [Hint: $$\ln (x) = \ln (x/a) + \ln (a)$$. This is a combination of functions which are continuous at $$a$$. Be sure to explain how you know that $$\ln (x/a)$$ is continuous at $$a$$.]

Q3

Write a formal deﬁnition of the statement $$f$$ is not continuous at $$a$$, and use it to prove that the function $$f(x) = \begin{cases} x & \text{ if } x\neq 1 \\ 0 & \text{ if } x= 1 \end{cases}$$ is not continuous at $$a = 1$$.

Contributor

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)