Speaking of the "challenge of modernist writing", we might be discussing 1) how modernist writers used stream-of-consciousness narrative and other techniques to present challenges to the reader in terms of cohesiveness, logical narrative flow, etc., or 2) how modernist writers took up the challenge of expressing the inner life and subjective experiences of characters in fiction.
This passage from Mrs. Dalloway reflects both of these challenges and takes place about two-thirds of the way into the novel.
"...it is the privilege of loneliness; in privacy one may do as one chooses. One might weep if no one saw. It had been his undoing - this susceptibility - in Anglo-Indian society; not weeping at the right time, or laughing either. I have that in me, he thought standing by the pillar-box, which could now dissolve in tears. Why, Heaven knows. Beauty of some sort probably, and the weight of the day, which beginning with that visit to Clarissa had exhausted him with its heat, its intensity, and the drip, drip, of one impression after another down into that cellar where they stood, deep, dark, and no one would ever know. Partly for that reason, its secrecy, complete and inviolable, he had found life like an unknown garden, full of turns and corners, surprising, yes; really it took one's breath away, these moments; there coming to him by the pillar box opposite the British Museum one of them, a moment in which things came together; this ambulance; and life and death."
In this passage, the past and present are considered almost simultaneously as the character attempts to make sense of a life and a world where impressions come "one after another" yet do not fully coalesce into a coherent unit of experience. Feelings and thoughts exist side by side in the prose, operating as a system of revelation and obscurity as the mind of the character attempts to understand itself and its environment, its history, and its nature.