Can a Continuous Bijection Have a Discontinuous Inverse?

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The discussion centers on finding a function f: E → R that is one-to-one, onto, and continuous, while having a discontinuous inverse f⁻¹: f(E) → R. A user suggests the exponential function e^x as a potential solution. Another participant requests a clearer presentation of the question, criticizing the use of distracting formatting. The conversation emphasizes the need for clarity in mathematical queries to facilitate better understanding and responses. The main focus remains on the properties of the function and its inverse in relation to continuity.
To0ta
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find : E\subseteqR

f : E\rightarrowR

1_1 , onto , contonuo

such that

f^{}-1 : f(E) \rightarrowR

is not continows

Please help me in finding a solution
 
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Can you repost it such that it's more readable please?
 
Perhaps you meant:

Let E\subseteq \mathbb{R}. Find a function f : E\rightarrow \mathbb{R} that is one-to-one, onto, and continuous such that f^{-1} : f(E) \rightarrow \mathbb{R} is not continuous.

Is this your question?
 
Yes, this is my question:smile:​
 
what about e^x?
 
To0ta said:
Yes, this is my question:smile:​


Why don't you just type your question? All of the stuff you are doing with special fonts, centering, and font size is distracting.
 

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