Problem with the boundaries of an integrable function

Question

I'm proving the following theorem: Let $\{K_n\}_{n=1}^{\infty}$ be a family of good kernel and $f$ and integrable function on the circle. Then: $$\lim_{n \to \infty} (f*K_n)(x) = f(x)$$ whenever f is continuous at x.
I wonder if an expression $$|f(x-y) - f(x)| \tag{1}$$ is bounded when $\delta<|y|<\pi$? Otherwise ($|y| < \delta$) I know that $|f(x-y) - f(x)| < \epsilon$ for a fixed $\epsilon$ because $f$ is continuous at x.
If $(1)$ is bounded I would appreciate an explanation.