Continuity Theorems: 18 flashcards
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1. State the Intermediate Value Theorem (IVT) precisely.
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If $f$ is continuous on $[a,b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c \in (a,b)$ such that $f(c) = k$.
2. What is the key hypothesis of the Intermediate Value Theorem?
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The function must be continuous on the closed interval $[a,b]$.
3. If $f$ is continuous on $[a,b]$ and $f(a)$ and $f(b)$ have opposite signs, what does the IVT guarantee?
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There exists at least one $c \in (a,b)$ such that $f(c) = 0$ (a root of the equation $f(x)=0$).
4. How do you use the IVT to prove that the equation $x^3 - x - 1 = 0$ has a root in $(1,2)$?
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Let $f(x)=x^3-x-1$. $f$ is a polynomial, hence continuous on $[1,2]$. Compute $f(1)=1-1-1=-1$ and $f(2)=8-2-1=5$. Since $f(1)<0$ and $f(2)>0$, by the IVT there exists $c \in (1,2)$ with $f(c)=0$.
5. What is the Extreme Value Theorem (EVT)?
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If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains both an absolute maximum and an absolute minimum value on $[a,b]$.
6. Why is the closed interval condition crucial for the Extreme Value Theorem?
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On an open interval or a non-closed interval, a continuous function may not attain its supremum or infimum (e.g., $f(x)=x$ on $(0,1)$ has no maximum).
7. State the definition of continuity at a point $x=a$.
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$f$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$. Equivalently, for every $\epsilon > 0$ there exists $\delta > 0$ such that if $|x-a|<\delta$, then $|f(x)-f(a)|<\epsilon$.
8. What are the three conditions required for $f$ to be continuous at $x=a$?
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1. $f(a)$ is defined. 2. $\lim_{x \to a} f(x)$ exists. 3. $\lim_{x \to a} f(x) = f(a)$.
9. If $f$ and $g$ are continuous at $x=a$, which combinations are also continuous at $x=a$?
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$f+g$, $f-g$, $f \cdot g$, and $f/g$ (provided $g(a) \neq 0$) are continuous at $x=a$.
10. How do you prove that a polynomial function is continuous everywhere?
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A polynomial is a sum of constant multiples of powers of $x$. Since $x$ is continuous and constant functions are continuous, and sums/products of continuous functions are continuous, every polynomial is continuous on $\mathbb{R}$.
11. What is the composition rule for continuity?
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If $g$ is continuous at $x=a$ and $f$ is continuous at $g(a)$, then the composite function $f \circ g$ is continuous at $x=a$.
12. Give an example of a function that is continuous on $(0,1)$ but does not attain a maximum value.
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$f(x) = x$ on $(0,1)$ is continuous but has no maximum (the supremum is 1, but 1 is not in the interval).
13. How can you use the IVT to show that a continuous function has a fixed point on $[0,1]$ if $f(0) \geq 0$ and $f(1) \leq 1$?
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Define $g(x) = f(x) - x$. Then $g(0) = f(0) - 0 \geq 0$ and $g(1) = f(1) - 1 \leq 0$. If $g(0)=0$ or $g(1)=0$, then 0 or 1 is a fixed point. Otherwise, $g(0)>0$ and $g(1)<0$, so by IVT there exists $c \in (0,1)$ with $g(c)=0$, i.e., $f(c)=c$.
14. What is the relationship between continuity and differentiability?
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If a function is differentiable at a point, then it is continuous at that point. The converse is not true (e.g., $f(x)=|x|$ is continuous but not differentiable at $x=0$).
15. What is a common strategy for proving that a function is continuous on an interval?
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Show that the function is continuous at every point in the interval. This can be done by using the definition (epsilon-delta), or by using the fact that sums, products, quotients, and compositions of continuous functions are continuous, and that polynomials, rational functions (where defined), and trigonometric functio…
16. How do you determine if a piecewise function is continuous at a point where the formula changes?
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Check the left-hand limit, right-hand limit, and the function value at that point. The function is continuous if all three are equal.
17. Give an example of a function that is continuous on $[0,1]$ but not differentiable at any point.
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The Weierstrass function is continuous everywhere but differentiable nowhere. (A simpler example: $f(x) = |x - 0.5|$ is continuous on $[0,1]$ but not differentiable at $x=0.5$.)
18. What is the 'sign change' method for finding roots of a continuous function?
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If $f$ is continuous on $[a,b]$ and $f(a)$ and $f(b)$ have opposite signs, then by the IVT there is at least one root in $(a,b)$. To approximate it, use the bisection method: repeatedly halve the interval, keeping the subinterval where the sign change occurs.
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