Two continuous functions on a closed interval guarantees a fixed point?
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Let $f,g: [0,1] to mathbb{R}$ be continuous functions and assume $f(0) > g(0)$ and $f(1) < g(1)$.
Prove there exists a $t in (0,1)$ such that $f(t) = g(t)$.
So far I have tried saying as f and g are continuous then f +g are continuous but how do I incorporate the inequality?
I understand the two functions must cross and therefore their meeting point would be t. I don't know which theorems I should use to prove this.
real-analysis analysis functions continuity
edited Nov 24 at 10:53
Viktor Glombik
469422
asked Nov 24 at 10:33
Pumpkinpeach
538
add a comment |
up vote
0
down vote
favorite
Let $f,g: [0,1] to mathbb{R}$ be continuous functions and assume $f(0) > g(0)$ and $f(1) < g(1)$.
Prove there exists a $t in (0,1)$ such that $f(t) = g(t)$.
So far I have tried saying as f and g are continuous then f +g are continuous but how do I incorporate the inequality?
I understand the two functions must cross and therefore their meeting point would be t. I don't know which theorems I should use to prove this.
real-analysis analysis functions continuity
edited Nov 24 at 10:53
Viktor Glombik
469422
asked Nov 24 at 10:33
Pumpkinpeach
538
@ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"...
– David C. Ullrich
Nov 24 at 16:09
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f,g: [0,1] to mathbb{R}$ be continuous functions and assume $f(0) > g(0)$ and $f(1) < g(1)$.
Prove there exists a $t in (0,1)$ such that $f(t) = g(t)$.
So far I have tried saying as f and g are continuous then f +g are continuous but how do I incorporate the inequality?
I understand the two functions must cross and therefore their meeting point would be t. I don't know which theorems I should use to prove this.
real-analysis analysis functions continuity
edited Nov 24 at 10:53
Viktor Glombik
469422
asked Nov 24 at 10:33
Pumpkinpeach
538
Let $f,g: [0,1] to mathbb{R}$ be continuous functions and assume $f(0) > g(0)$ and $f(1) < g(1)$.
Prove there exists a $t in (0,1)$ such that $f(t) = g(t)$.
So far I have tried saying as f and g are continuous then f +g are continuous but how do I incorporate the inequality?
I understand the two functions must cross and therefore their meeting point would be t. I don't know which theorems I should use to prove this.
real-analysis analysis functions continuity
real-analysis analysis functions continuity
edited Nov 24 at 10:53
Viktor Glombik
469422
asked Nov 24 at 10:33
Pumpkinpeach
538
edited Nov 24 at 10:53
Viktor Glombik
469422
asked Nov 24 at 10:33
Pumpkinpeach
538
edited Nov 24 at 10:53
Viktor Glombik
469422
edited Nov 24 at 10:53
Viktor Glombik
469422
edited Nov 24 at 10:53
Viktor Glombik
469422
469422
asked Nov 24 at 10:33
Pumpkinpeach
538
asked Nov 24 at 10:33
Pumpkinpeach
538
asked Nov 24 at 10:33
Pumpkinpeach
538
538
@ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"...
– David C. Ullrich
Nov 24 at 16:09
add a comment |
@ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"...
– David C. Ullrich
Nov 24 at 16:09
@ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"...
– David C. Ullrich
Nov 24 at 16:09
@ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"...
– David C. Ullrich
Nov 24 at 16:09
add a comment |
2 Answers
2
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2
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In exercises where you have continuity, equality, and want to prove that there exists an element that satisfies a certain equation, think about the intermediate value theorem.
Hint: Proving that $f(t)=g(t)$ for some $t$ means that $f(t)-g(t)=0$ for some $t$, so consider the function $h=f-g$, which is continuous.
answered Nov 24 at 10:41
Scientifica
6,34141333
add a comment |
up vote
2
down vote
Hint:Let $h(x)=f(x)-g(x)$ be a continuous function. Since $h(0) gt 0$ and $h(1) lt 0$, $h(x)$ must be 0 somewhere in the interval.
answered Nov 24 at 10:41
Thomas Shelby
1,195116
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
In exercises where you have continuity, equality, and want to prove that there exists an element that satisfies a certain equation, think about the intermediate value theorem.
Hint: Proving that $f(t)=g(t)$ for some $t$ means that $f(t)-g(t)=0$ for some $t$, so consider the function $h=f-g$, which is continuous.
answered Nov 24 at 10:41
Scientifica
6,34141333
add a comment |
up vote
2
down vote
In exercises where you have continuity, equality, and want to prove that there exists an element that satisfies a certain equation, think about the intermediate value theorem.
Hint: Proving that $f(t)=g(t)$ for some $t$ means that $f(t)-g(t)=0$ for some $t$, so consider the function $h=f-g$, which is continuous.
answered Nov 24 at 10:41
Scientifica
6,34141333
add a comment |
up vote
2
down vote
up vote
2
down vote
In exercises where you have continuity, equality, and want to prove that there exists an element that satisfies a certain equation, think about the intermediate value theorem.
Hint: Proving that $f(t)=g(t)$ for some $t$ means that $f(t)-g(t)=0$ for some $t$, so consider the function $h=f-g$, which is continuous.
answered Nov 24 at 10:41
Scientifica
6,34141333
In exercises where you have continuity, equality, and want to prove that there exists an element that satisfies a certain equation, think about the intermediate value theorem.
Hint: Proving that $f(t)=g(t)$ for some $t$ means that $f(t)-g(t)=0$ for some $t$, so consider the function $h=f-g$, which is continuous.
answered Nov 24 at 10:41
Scientifica
6,34141333
answered Nov 24 at 10:41
Scientifica
6,34141333
answered Nov 24 at 10:41
Scientifica
6,34141333
answered Nov 24 at 10:41
Scientifica
6,34141333
6,34141333
add a comment |
add a comment |
up vote
2
down vote
Hint:Let $h(x)=f(x)-g(x)$ be a continuous function. Since $h(0) gt 0$ and $h(1) lt 0$, $h(x)$ must be 0 somewhere in the interval.
answered Nov 24 at 10:41
Thomas Shelby
1,195116
add a comment |
up vote
2
down vote
Hint:Let $h(x)=f(x)-g(x)$ be a continuous function. Since $h(0) gt 0$ and $h(1) lt 0$, $h(x)$ must be 0 somewhere in the interval.
answered Nov 24 at 10:41
Thomas Shelby
1,195116
add a comment |
up vote
2
down vote
up vote
2
down vote
Hint:Let $h(x)=f(x)-g(x)$ be a continuous function. Since $h(0) gt 0$ and $h(1) lt 0$, $h(x)$ must be 0 somewhere in the interval.
answered Nov 24 at 10:41
Thomas Shelby
1,195116
Hint:Let $h(x)=f(x)-g(x)$ be a continuous function. Since $h(0) gt 0$ and $h(1) lt 0$, $h(x)$ must be 0 somewhere in the interval.
answered Nov 24 at 10:41
Thomas Shelby
1,195116
edited Nov 25 at 13:17
answered Nov 24 at 10:41
Thomas Shelby
1,195116
answered Nov 24 at 10:41
Thomas Shelby
1,195116
answered Nov 24 at 10:41
Thomas Shelby
1,195116
1,195116
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@ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"...
– David C. Ullrich
Nov 24 at 16:09