Calculus I - Continuity

14 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Continuity at a point aaa

   A function f(x)f(x)f(x) is continuous at aaa if lim⁡x→af(x)=f(a)\lim_{x\to a}f(x)=f(a)limx→a​f(x)=f(a)

2. Checklist for continuity

   f(a)f(a)f(a) is defined lim⁡x→af(x)\lim_{x\to a}f(x)limx→a​f(x) exists lim⁡x→af(x)=f(a)\lim_{x\to a}f(x)=f(a)limx→a​f(x)=f(a)

3. Removable discontinuity

   A discontinuity where lim⁡x→af(x)≠f(a)\lim_{x\to a}f(x)\neq f(a)limx→a​f(x)=f(a) , which can be removed by redefining f(x)f(x)f(x) at aaa to be lim⁡x→af(x)\lim_{x\to a}f(x)limx→a​f(x)

4. Jump discontinuity

   A discontinuity where lim⁡x→af(x)\lim_{x\to a}f(x)limx→a​f(x) does not exist, due to lim⁡x→a+f(x)≠lim⁡x→a−f(x)\lim_{x\to a^+}f(x)\neq \lim_{x\to a^-}f(x)limx→a+​f(x)=limx→a−​f(x) (the function looks like it jumps in value)

5. Infinite discontinuity

   A discontinuity where lim⁡x→af(x)\lim_{x\to a}f(x)limx→a​f(x) does not exist, due to one or both of the one sided limits being an infinity ( ∞∉R\infty \notin \mathbb{R}∞∈/R )

6. Continuity on an interval

   A function f(x)f(x)f(x) defined on an interval III is continuous on the III if f(x)f(x)f(x) is continuous at every point on the interior of III and: If aaa is the left-hand (negative-side) endpoint contained in III , lim⁡x→a+f(x)=f(a)\lim_{x\to a^+}f(x)=f(a)limx→a+​f(x)=f(a) If bbb is the right-hand (positive-side) endpoint contained in III , lim⁡x→b−f(x)=f(b)\lim_{x\to b^-}f(x)=f(b)limx→b−​f(x)=f(b) (A.K.A f(x)f(x)f(x) is continuous on III from the right and the left)

7. All of the following functions are continuous on the intervals where they are defined

   Polynomials Rational functions (except where the denominator = 0) Exponential functions Logarithmic functions Trigonometric functions Inverse trigonometric functions

8. If fff and ggg are continuous at aaa , then the following functions are also continuous at aaa . Assume c∈Rc \in \mathbb{R}c∈R and 0"> n∈Z ∧ n>0n \in \mathbb{Z} \text{ }\wedge \text{ } n>0n∈Z ∧ n>0

   f+gf + gf+g f−gf-gf−g cfcfcf fgfgfg fg\frac{f}{g}gf​ , assuming g(a)≠0g(a) \neq 0g(a)=0 (f(x))n(f(x))^n(f(x))n f∘gf \circ gf∘g

9. Assume 0"> n∈Z ∧ n>0n \in \mathbb{Z} \text{ } \wedge \text{ } n>0n∈Z ∧ n>0 . When is (f(x))1n(f(x))^{\frac{1}{n}}(f(x))n1​ continuous?

   If nnn is odd then (f(x))1n(f(x))^{\frac{1}{n}}(f(x))n1​ is continuous at all points at which fff is continuous. If nnn is even then (f(x))1n(f(x))^{\frac{1}{n}}(f(x))n1​ is continuous at at all points aaa at which fff is continuous and 0"> f(a)>0f(a) > 0f(a)>0

10. Intermediate Value Theorem (IVT)

    Suppose fff is continuous on the interval [a,b][a, b][a,b] and LLL is a number strictly between f(a)f(a)f(a) and f(b)f(b)f(b) . There exists at least one number ccc in (a,b)(a, b)(a,b) satisfying f(c)=Lf(c)=Lf(c)=L

11. Rolle's Theorem

    Suppose fff is a function defined on a closed interval [a,b][a,b][a,b] . If: fff is continuous on [a,b][a,b][a,b] fff is differentiable on (a,b)(a,b)(a,b) f(a)=f(b)f(a)=f(b)f(a)=f(b) Then there exists at least one number ccc in the open interval (a,b)(a,b)(a,b) for which f′(c)=0f'(c)=0f′(c)=0

12. Mean Value Theorem (MVT)

    Let fff be a function defined on a closed interval [a,b][a,b][a,b] If: fff is continuous on [a,b][a,b][a,b] fff is differentiable on (a,b)(a,b)(a,b) Then there is at least one number ccc in the open interval (a,b)(a,b)(a,b) for which f′(c)=f(b)−f(a)b−af'(c)=\frac{f(b)-f(a)}{b-a}f′(c)=b−af(b)−f(a)​ , therefore implying there exists some secant line formed on (a,b)(a,b)(a,b) that has the same slope as the derivative formed at f′(c)f'(c)f′(c)

13. Constant corollary of the MVT

    If a function fff is continuous on the closed interval [a,b][a,b][a,b] and is differentiable on the open interval (a,b)(a,b)(a,b) , and if f′(x)=0f'(x)=0f′(x)=0 for all numbers xxx in (a,b)(a,b)(a,b) , then fff is constant on (a,b)(a,b)(a,b) . Relating to rectilinear motion, if an object in rectilinear motion has a velocity v=0v=0v=0 , then the object is at rest

14. Same derivative corollary of the MVT

    If the functions fff and ggg are differentiable on an open interval (a,b)(a,b)(a,b) and if f′(x)=g′(x)f'(x)=g'(x)f′(x)=g′(x) for all numbers xxx in (a,b)(a,b)(a,b) , then there exists some number CCC for which f(x)=g(x)+Cf(x)=g(x)+Cf(x)=g(x)+C (if the functions have the same derivative over an interval, they only differ by a constant)