Understanding proof that continuous image of connected is connected

general-topology proof-explanation connectedness

Asserting that the image of $f$ is connected is asserting that $f(X)$ is connected. What happens in $Y\setminus f(X)$ does not matter. So, it is natural to assume from the start that $Y$ is just $f(X)$; in other words, that $f$ is surjective.

Related

Least Square Estimation of Linear Regression
How is it that $\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$ for all angles, even though the derivation holds only for $\cos A\cos B\neq 0$?
What's the measure of the $\angle POB$ in the circle below?
Limit of the variance of a difference of indicators
Show that $O(n,\mathbb{R})$ the set of all orthogonal matrices is closed in $M(n,\mathbb{R})$ [duplicate]
How to show an inequality in an inner product space?
Limit with sequence of functions and integrals
Let $f(x) = x^3-x^2–3x–1$ and $h(x)=\frac{f(x)}{g(x)}$ where $h(x)$ is a rational function such that
Is the Dihedral Group of order $24$ isomorphic to the Symmetric Group on $4$ elements? [closed]
Variation in density of the Farey sequence in the interval $[0,1]$
Is it possible to retrieve a list of all users that liked a tweet?
Calculating $\int x dx$ using trigonometric functions