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I'm trying to show $\tanh(x)$ is bijective using the intermediate value theorem.

It works by noting $\tanh(x)$ as strictly increasing by differentiating $\tanh(x)$ and then surjective using limits to $\infty$ and $-\infty$.

However, the intermediate value theorem (to prove surjectivity) needs a continuous function. Where/how does one show that $\tanh(x)$ is continuous?

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$\tanh$ is defined to be $$ \tanh = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^{2x} - 1}{e^{2x} + 1} $$ Since $\tanh$ is the quotient of two continuous functions, and the denominator is never $0$, $\tanh$ is continuous.

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