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How to construct continuous function between sequences properties of orderd upper and lower semi continuous functions

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.

If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it true that $f$ is continuous at $x_0$? What (I think it is true just using the definition of semi-continuous functions using $\limsup_{x\rightarrow y} f(x) \leq f(y)$ and $\liminf_{x\rightarrow y} g(x) \geq g(y)$)

What about the topology of the set of the continuous points of $f$? Or the topology of the set of $x$ such that $f(x) = g(x)$ (open or closed?)

Thank you!

How to construct continuous function between sequences of upper and lower semi continuous functions

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.

If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it true that $f$ is continuous at $x_0$? What about the topology of the set of the continuous points of $f$? (open or closed?)

Thank you!

properties of orderd upper and lower semi continuous functions

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.

If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it true that $f$ is continuous at $x_0$? (I think it is true just using the definition of semi-continuous functions using $\limsup_{x\rightarrow y} f(x) \leq f(y)$ and $\liminf_{x\rightarrow y} g(x) \geq g(y)$)

What about the topology of the set of the continuous points of $f$? Or the topology of the set of $x$ such that $f(x) = g(x)$ (open or closed?)

Thank you!

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Let $x\in X$ which$M$ is a compact space (for example $S^1$). Suppose Assume $\{f_n(x)\}_{n\geq1}$$f$ is upper semi-continuous on $M$, decreasing and $\{g_n(x)\}_{n\geq}$$g$ is lower semi-continuous, increasing on $M$, and $f_n(x)\geq g_n(x)$$f(x) \geq g(x)$ for any $x\in M$.

If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it possible to construct a continuous function $h(x)$ in between,true that $f$ is continuous at $f_n(x) \geq h(x)\geq g_n(x)$$x_0$? What about the topology of the set of the continuous points of $f$? (open or closed?)

Thank you!

Let $x\in X$ which is compact space (for example $S^1$). Suppose $\{f_n(x)\}_{n\geq1}$ is upper semi-continuous, decreasing and $\{g_n(x)\}_{n\geq}$ is lower semi-continuous, increasing, and $f_n(x)\geq g_n(x)$, is it possible to construct a continuous function $h(x)$ in between, that is $f_n(x) \geq h(x)\geq g_n(x)$?

Thank you!

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.

If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it true that $f$ is continuous at $x_0$? What about the topology of the set of the continuous points of $f$? (open or closed?)

Thank you!

Post Closed as "Not suitable for this site" by Pietro Majer, Wolfgang, Alexey Ustinov, Alex Degtyarev, Stefan Kohl
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How to construct continuous function between sequences of upper and lower semi continuous functions

Let $x\in X$ which is compact space (for example $S^1$). Suppose $\{f_n(x)\}_{n\geq1}$ is upper semi-continuous, decreasing and $\{g_n(x)\}_{n\geq}$ is lower semi-continuous, increasing, and $f_n(x)\geq g_n(x)$, is it possible to construct a continuous function $h(x)$ in between, that is $f_n(x) \geq h(x)\geq g_n(x)$?

Thank you!