Continuous effects are effects that neither start a chain nor continue a chain. They are passive effects that will continue to affect the field while it remains face-up meaning they can activate during chain's if they're condition(s) are fulfilled.

Continuous effects do not have effects until their summon is successful before their effect takes place.

Ex. Jinzo can be negated and destroyed by Solemn Judgement.

Continuous effects are able to bypass Spell Speed 3 spells if they are on the field first.

Ex. Jinzo can prevent Solemn Judgment from activating.

Continuous effects are NEVER optional!

How to identify Continuous effects
There are many forms of texts for continuous effects, but they're usually worded with "While" or "if this card". The most common is stating that the monster is or has to be "Face-Up".

Confusion
Trigger Effects are also associated with "If this card", however they are optional. Keep in mind if the word "If" and "you can" are in the same sentence, it is not continuous.

Credits to Sinisterplantboy:
If a card text does not have a colon ( : ) or semi-colon ( ; ), it does not start a chain.

Replacement Effects
Cards with replacement effects like Gachi Gachi Gantetsu's "detach effect" do not start a chain; they are continuous. This is the only type of effect that bypasses the "optional rule" for continuous effects.

Cards with "If this face-up card would be _____, you can _____ instead." do not start chains.

Ex. of a Continuous Effect and Replacement Effect