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Through galleries of graphs and short filmstrips, this paper aims to sharpen students' eyes for visually recognizing continuous functions. It seeks to develop intuition for what visual features of a graph continuity does and does not allow for. We have found that even students who can work correctly with rigorous definitions may not be able to recognize a continuous function from its graph. To help mitigate this problem, these visuals can be used alongside the definitions in analysis courses, and in calculus courses where the same definitions are used. The content is arranged into a series of pictures accompanied by questions, with suggestions for the instructor on how to guide students in their learning. The design allows students to make their own discoveries and connections. The investigations crescendo to a surprising fact: that almost all continuous functions are differentiable nowhere!

GluA1, GluA2, GluA3, and GluA4 are the constitutive subunits of amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPARs), the major mediators of fast excitatory transmission in the mammalian central nervous system. Most AMPARs are Ca(2+)-impermeable because of the presence of the GluA2 subunit. GluA2 mRNA undergoes an editing process that results in a Q-R substitution, a key factor in the regulation of AMPAR Ca(2+)-permeability. AMPARs lacking GluA2 or containing the unedited subunit are permeable to Ca(2+) and Zn(2+). The phenomenon physiologically modulates synaptic plasticity while, in pathologic conditions, leads to increased vulnerability to excitotoxic neuronal death. Given the importance of these subunits, we have therefore evaluated possible associations between changes in expression levels of AMPAR subunits and development of cognitive deficits in 3xTg-AD mice, a widely investigated transgenic mouse model of Alzheimer's disease (AD). With quantitative real-time PCR analysis, we assayed hippocampal mRNA expression levels of GluA1-4 subunits occurring in young [3 months of age (m.o.a.)] and old (12 m.o.a) Tg-AD mice and made comparisons with levels found in age-matched wild type (WT) mice. Efficiency of GluA2 RNA editing was also analyzed. All animals were cognitively tested for learning short- and long-term spatial memory with the Morris Water Maze (MWM) navigation task. 3xTg-AD mice showed age-dependent decreases of mRNA levels for all the AMPAR subunits, with the exception of GluA2. Editing remained fully efficient with aging in 3xTg-AD and WT mice. A one-to-one correlation analysis between MWM performances and GluA1-4 mRNA expression profiles showed negative correlations between GluA2 levels and MWM performances in young 3xTg-AD mice. On the contrary, positive correlations between GluA2 mRNA and MWM performances were found in young WT mice. Our data suggest that increases of AMPARs that contain GluA1, GluA3, and GluA4 subunits may help in maintaining cognition in pre-symptomatic 3xTg-AD mice.

mRNA chimeras from chromosomal translocations often play a role as transforming oncogenes. However, cancer transcriptomes also contain mRNA chimeras that may play a role in tumor development, which arise as transcriptional or post-transcriptional events. To identify such chimeras, we developed a deterministic screening strategy for long-range sequence analysis. High-throughput, long-read sequencing was then performed on cDNA libraries from major tumor histotypes and corresponding normal tissues. These analyses led to the identification of 378 chimeras, with an unexpectedly high frequency of expression (≈2 x 10-5 of all mRNA). Functional assays in breast and ovarian cancer cell lines showed that a large fraction of mRNA chimeras regulates cell replication. Strikingly, chimeras were shown to include both positive and negative regulators of cell growth, which functioned as such in a cell-type-specific manner. Replication-controlling chimeras were found to be expressed by most cancers from breast, ovary, colon, uterus, kidney, lung, and stomach, suggesting a widespread role in tumor development.

mRNA chimeras from chromosomal translocations often play a role as transforming oncogenes. However, cancer transcriptomes also contain mRNA chimeras that may play a role in tumor development, which arise as transcriptional or post-transcriptional events. To identify such chimeras, we developed a deterministic screening strategy for long-range sequence analysis. High-throughput, long-read sequencing was then performed on cDNA libraries from major tumor histotypes and corresponding normal tissues. These analyses led to the identification of 378 chimeras, with an unexpectedly high frequency of expression (≈2 x 10(-5) of all mRNA). Functional assays in breast and ovarian cancer cell lines showed that a large fraction of mRNA chimeras regulates cell replication. Strikingly, chimeras were shown to include both positive and negative regulators of cell growth, which functioned as such in a cell-type-specific manner. Replication-controlling chimeras were found to be expressed by most cancers from breast, ovary, colon, uterus, kidney, lung, and stomach, suggesting a widespread role in tumor development.