For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
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Calculus Volume 1 · 2.4 Continuity
Continuity: practice problems
25 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.
Exercises 133
Exercises 135
Exercises 137
Exercises 139
Exercises 141
Exercises 143
Exercises 145
Exercises 147
Exercises 149
Exercises 151
Exercises 153
Exercises 155
Exercises 157
Exercises 159
Exercises 161
Exercises 163
Exercises 165
Exercises 167
Exercises 169
Exercises 171
Exercises 173
Exercises 175
Review Exercises 211
Review Exercises 227
Key terms · 9
- continuity at a point
- A function $f (x)$ is continuous at a point a if and only if the following three conditions are satisfied: (1) $f (a)$ is defined, (2) $\lim_{x \to a} f (x)$ exists, and (3) $\lim_{x \to a} f (x) = f (a)$
- continuity from the left
- A function is continuous from the left at b if $\lim_{x \to b^-} f (x) = f (b)$
- continuity from the right
- A function is continuous from the right at a if $\lim_{x \to a^+} f (x) = f (a)$
- continuity over an interval
- a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function $f (x)$ is continuous over a closed interval of the form $[a, b]$ if it is continuous at every point in $(a, b),$ and it is continuous from the right at a and from the left at b
- discontinuity at a point
- A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point
- infinite discontinuity
- An infinite discontinuity occurs at a point a if $\lim_{x \to a^-} f (x) = \pm \infty$ or $\lim_{x \to a^+} f (x) = \pm \infty$
- Intermediate Value Theorem
- Let f be continuous over a closed bounded interval $[\text{a}, \text{b}];$ if z is any real number between $f (a)$ and $f (b),$ then there is a number c in $[a, b]$ satisfying $f (c) = z$
- jump discontinuity
- A jump discontinuity occurs at a point a if $\lim_{x \to a^-} f (x)$ and $\lim_{x \to a^+} f (x)$ both exist, but $\lim_{x \to a^-} f (x) \ne \lim_{x \to a^+} f (x)$
- removable discontinuity
- A removable discontinuity occurs at a point a if $f (x)$ is discontinuous at a, but $\lim_{x \to a} f (x)$ exists
Learn 2.4 on the whiteboard
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