Where is {{MATH_153}} continuous? {{CMD_154}} {{CMD_155}} {{CMD_156}}for all real numbers{{CMD_157}} {{CMD_158}}at {{MATH_159}}{{CMD_160}} {{CMD_161}}for all real numbers, except {{MATH_162}}{{CMD_163}} {{CMD_164}}impossible to say{{CMD_165}} {{CMD_166}} {{CMD_167}} {{CMD_168}} {{CMD_169}} True or false: If {{MATH_170}} and {{MATH_171}} are continuous functions on an interval {{MATH_172}}, then {{MATH_173}} is continuous on {{MATH_174}}. {{CMD_175}} {{CMD_176}} {{CMD_177}}True{{CMD_178}} {{CMD_179}}False{{CMD_180}} {{CMD_181}} {{CMD_182}} Let's assume that {{MATH_183}} is an open interval and {{MATH_184}} is a number in {{MATH_185}}. Remember, since {{MATH_186}} and {{MATH_187}} are both continuous on {{MATH_188}}, they are both continuous at {{MATH_189}}. This means that {{MATH_190}} and {{MATH_191}}. Now, define a new function, {{MATH_192}}, where {{MATH_193}}, for all {{MATH_194}} in {{MATH_195}}. We have to show that {{MATH_196}} is continuous at {{MATH_197}}, or that {{MATH_198}} Lat's start with {{MATH_199}} and, therefore, {{MATH_200}} We have proved that {{MATH_201}} is continuous at any number {{MATH_202}} in {{MATH_203}}. Therefore, {{MATH_204}} is continuous on {{MATH_205}}. Similarly, we can prove that {{MATH_206}} is continuous on any interval {{MATH_207}}, by showing it is left-or right-continuous at the endpoints. We can adjust the proof for the function {{MATH_208}}. {{CMD_209}} {{CMD_210}} {{CMD_211}}