Is $\frac{x}{x-1}$ continuous on $(1,\infty)$? Prove it.

Looking at the graph of the function, I assumed that it is continuous. I have tried to prove it by the $\epsilon$-$\delta$ definition of continuity as follows-

For any $x,c \in (1,\infty)$ $$ 0 < |x-c| < \delta $$ We have to show that $$ \left |\frac{x}{x-1} - \frac{c}{c-1} \right | < \epsilon$$ We can write the LHS as $$ \left |\frac{x-c}{(x-1)(c-1)} \right | < \frac{\delta}{|x-1||c-1|} $$ However I am stuck in trying to choose a $\delta$ dependent on $c$ and $\epsilon$ such that the above expression is less than $\epsilon$. The main problem I am facing is that as x can be very close to $1$, the fraction will explode. I am a novice at continuity proofs so I would appreciate detailed answers. Thanks in advance

Hint: $|x-1| \geq c-1 -|x-c|>c-1-\delta >\frac {c-1} 2$ if $0<\delta <\frac {c-1} 2$. Can you finish?

[$0 <\delta <\min \{{\frac {c-1} 2, \frac {2\epsilon} {(c-1)^{2}}}\}$ would do].