# Thread:

1. ## inequalities

Use the inequality absolute value[sin(a)-sin(b)] << absolute value[a-b], which is valid for all real numbers a and b, to prove that sin x is continuous on R.

2. just apply the mean value theorem for the function $f(t)=\sin t$ on the interval $[a,b].$

3. Is the mean value theorem the same thing as the intermediate value theorem?

IVT
f(x) is defined on [a,b]
f(x) assumes all values of (a,b)
then f(c) where c is a point in (a,b) intersects f(x)

4. no, the assumptions are different, google it and see.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8301753401756287, "perplexity_flag": "middle"}