what is the largest $\delta >0$ which fits the definition of continuity of the function

Question

Given $\epsilon >0$, what is the largest $\delta >0$ which fits the definition of continuity of the function

$$f(x)=\begin{cases} \frac{x+3}{2},& \text{if } x \leq 1\\ \frac{7-x}{3}, &\text{if } x\geq 1\\ \end{cases}$$

at $x=1.$

I try to find out $\vert f(x)-f(1)\vert < \epsilon$ for two different cases but I get nothing. Please help.

Answer 1

For $x<1$: $|f(x)-f(1)|=|\frac{x+3}{2}-2|=\frac{1}{2}|x-1|$

and

for $x<1$: $|f(x)-f(1)|=|\frac{7-x}{3}-2|=\frac{1}{3}|x-1|$.

Can you proceed ?